Group theory

Group theory is a branch of mathematics that studies algebraic structures known as groups. A group is defined as a set equipped with a single binary operation that satisfies four fundamental properties: 1. **Closure**: If \( a \) and \( b \) are elements of the group, then the result of the operation \( a * b \) is also in the group.

Abelian group theory is a branch of abstract algebra that focuses on the study of Abelian groups (or commutative groups). An **Abelian group** is a set equipped with an operation that satisfies certain properties: 1. **Closure**: For any two elements \( a \) and \( b \) in the group, the result of the operation (usually denoted as \( a + b \) or \( ab \)) is also in the group.

An **Abelian Lie group** is a type of Lie group in which the group operation is commutative. This means that for any two elements \( g \) and \( h \) in the group \( G \), the following property holds: \[ g \cdot h = h \cdot g \] where \( \cdot \) represents the group operation.

An Abelian group, also known as a commutative group, is a set equipped with a binary operation that satisfies certain properties. Specifically, a group \((G, *)\) is called Abelian if it satisfies the following criteria: 1. **Closure**: For all \(a, b \in G\), the result of the operation \(a * b\) is also in \(G\).

An "algebraically compact group" is a concept primarily found in the context of algebraic groups, a subject at the intersection of algebra and geometry. In broad terms, an **algebraic group** is a group that is also an algebraic variety, meaning it can be described by polynomial equations. These groups arise in various branches of mathematics, including number theory, algebraic geometry, and representation theory.

The Baer–Specker group, often denoted as \( BS \), is a classical example in the field of group theory, specifically in the study of torsion-free abelian groups. It is an important structure for various reasons, including its role in representation theory and its properties as a divisible group.

In group theory, a branch of abstract algebra, a **basic subgroup** typically refers to a subgroup that exhibits certain essential properties in the context of finite group theory, particularly in relation to p-groups and the Sylow theorems. However, it's important to clarify that the term "basic subgroup" is not standard across all texts and contexts and can have specific meanings depending on the area of interest.

The term "Butler Group" could refer to a few different things, depending on the context. One prominent reference is to the Butler Group in the context of technology and research. The Butler Group was a well-known IT research and advisory firm that provided insights into emerging technologies, trends, and market analysis for businesses. They focused on helping organizations understand and leverage technology effectively.

In the context of module theory, a **cotorsion group** refers to an abelian group (or more generally, a module) where every element is "cotorsion" in a certain sense.

A cyclic group is a type of group in which every element can be expressed as a power (or multiple) of a single element, known as a generator. In more formal terms, a group \( G \) is called cyclic if there exists an element \( g \in G \) such that every element \( a \in G \) can be written as \( g^n \) for some integer \( n \).

In the context of group theory, a **divisible group** is a particular type of abelian group (a group where the group operation is commutative) that satisfies a specific divisibility condition related to its elements.

An **elementary abelian group** is a specific type of group that is both abelian (commutative) and has a particular structure in which every non-identity element has an order of 2. This means that for every element \( g \) in the group, if \( g \neq e \) (where \( e \) is the identity element of the group), then \( g^2 = e \).

In group theory, a branch of abstract algebra, an essential subgroup is a specific type of subgroup that has particular relevance in the context of group actions and the structure of groups. A subgroup \( H \) of a group \( G \) is said to be essential in \( G \) if it intersects every nontrivial subgroup of \( G \).

The group of rational points on the unit circle refers to the set of points \( (x, y) \) on the unit circle defined by the equation \[ x^2 + y^2 = 1 \] where both \( x \) and \( y \) are rational numbers (numbers that can be expressed as fractions of integers). To describe the rational points on the unit circle, we can parameterize the unit circle using trigonometric functions or with rational parameterization.

In the context of abelian groups, the term "height" can refer to a couple of different concepts depending on the specific area of mathematics being considered, such as group theory or algebraic geometry. 1. **In Group Theory**: The height of an abelian group can refer to a measure of the complexity of the group, particularly when it comes to finitely generated abelian groups.

The Herbrand quotient is a concept from model theory and mathematical logic, particularly within the context of the study of formal systems and the properties of logical formulas. It generally pertains to measuring certain aspects of structures in formal theories, especially in relation to the notion of definability and algebraic properties of models. Specifically, the Herbrand quotient is defined in the context of Herbrand's theorem, which relates to the concept of Herbrand universes and Herbrand bases.

A **locally compact abelian group** is a type of mathematical structure that combines concepts from both topology and group theory. Here's a breakdown of what this term means: 1. **Group**: In mathematics, a group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverse elements.

In group theory, a **locally cyclic group** is a type of group that is, in a certain sense, generated by its own elements in a cyclic manner. More formally, a group \( G \) is said to be locally cyclic if every finitely generated subgroup of \( G \) is cyclic. This means that for any finite set of elements from \( G \), the subgroup generated by those elements can be generated by a single element.

In the context of abelian groups, the term "norm" can refer to a couple of different concepts depending on the specific field of mathematics being discussed. One common usage, particularly in algebra and number theory, is the notion of a norm associated with a field extension or a number field.

A primary cyclic group is a specific type of cyclic group in the field of group theory, a branch of abstract algebra. A cyclic group is one that can be generated by a single element, meaning that every element of the group can be expressed as a power (or multiple) of this generator.

A Prüfer group, also known as a Prüfer \(p\)-group, is a type of abelian group that can be defined for a prime number \(p\).

Prüfer's Theorem refers to a couple of important results in the context of graph theory, particularly regarding trees. Here are the two main aspects of Prüfer's Theorem often discussed: 1. **Prüfer Code (or Prüfer Sequence)**: The theorem states that there is a one-to-one correspondence between labeled trees with \( n \) vertices and sequences of length \( n-2 \) made up of labels from \( 1 \) to \( n \).

In group theory, a "pure subgroup" refers to a specific type of subgroup within an abelian group. Specifically, a subgroup \( H \) of an abelian group \( G \) is called a **pure subgroup** if it satisfies a certain property concerning integer multiples.

The **rank** of an abelian group is a concept that generally refers to the maximum number of linearly independent elements in the group when it is considered as a module over the integers. For finitely generated abelian groups, the rank can be understood in relation to the structure theorem for finitely generated abelian groups.

A **topological abelian group** is a mathematical structure that combines the concepts of a group and a topology. Specifically, it is an abelian group that has a compatible topology, allowing for the notions of continuity and convergence to be defined in the context of group operations.

In the context of group theory, particularly in the study of abelian groups (and more generally, in the context of modules over a ring), the **torsion subgroup** is an important concept. The torsion subgroup of an abelian group \( G \) is defined as the set of elements in \( G \) that have finite order.

Algebraic groups are a central concept in an area of mathematics that blends algebra, geometry, and number theory. An algebraic group is defined as a group that is also an algebraic variety, meaning that its group operations (multiplication and inversion) can be described by polynomial equations. More formally, an algebraic group is a set that satisfies the group axioms (associativity, identity, and inverses) and is also equipped with a structure of an algebraic variety.

An Abelian variety is a special type of algebraic variety that is defined over a field, typically the field of complex numbers or a finite field. They have a number of important properties that make them central to the study of algebraic geometry and number theory. Here are some key characteristics and definitions related to Abelian varieties: 1. **Group Structure**: An Abelian variety is not just a geometric object; it has a natural structure that turns it into a group.

Algebraic homogeneous spaces are mathematical structures that arise in the context of algebraic geometry and representation theory. More specifically, they are typically associated with algebraic groups and their actions on varieties. ### Definition An **algebraic homogeneous space** can be defined in the following way: 1. **Algebraic Group**: Let \( G \) be an algebraic group defined over an algebraically closed field (like the field of complex numbers).

E8 is a highly complex and deeply interesting mathematical structure that appears in various areas, including geometry, algebra, and theoretical physics. It is most commonly referred to in the context of group theory and is one of the five exceptional simple Lie groups. Here are some key points about E8: 1. **Lie Group**: E8 is one of the simplest types of continuous symmetry groups, known as a Lie group. Simple Lie groups are those that cannot be decomposed into smaller, simpler groups.

Exceptional Lie algebras are a special class of Lie algebras that are neither classical nor affine. They are characterized by their exceptional properties, most notably their dimension and the structure of their root systems. Unlike the classical Lie algebras (which include types A, B, C, D corresponding to the classical groups, and E, F, G corresponding to exceptional types), the exceptional Lie algebras cannot be directly described in terms of standard matrix groups.

Linear algebraic groups are a fundamental concept in the field of algebraic geometry that connect algebraic groups and linear algebra. More specifically, a linear algebraic group is a group that is also an algebraic variety, where the group operations (multiplication and inversion) are given by polynomial functions.

Representation theory of algebraic groups is a branch of mathematics that studies how algebraic groups can act on vector spaces through linear transformations. More specifically, it examines the ways in which algebraic groups can be represented as groups of matrices, and how these representations can be understood and classified. ### Key Concepts: 1. **Algebraic Groups**: These are groups that have a structure of algebraic varieties.

An **adelic algebraic group** is a concept that arises in the context of algebraic groups and number theory, particularly in the study of rational points and arithmetic geometry. To explain it more precisely, we first need to understand what an algebraic group is and then what "adelic" means in this context. ### Algebraic Groups An **algebraic group** is a group that is also an algebraic variety.

An **algebraic group** is a group that is also an algebraic variety, where the group operations (multiplication and taking inverses) are given by polynomial functions. More formally, an algebraic group is a set equipped with a group structure and additional structure that satisfies certain properties of being defined over algebraically closed fields. ### Key Concepts 1. **Algebraic Variety**: An algebraic variety is a geometric object defined as the solution set of a system of polynomial equations.

In the context of algebraic groups, approximation often refers to various ways to understand and study algebraic structures through simpler or more manageable models. The term could encompass different specific concepts depending on the branch of mathematics or the particular problems being addressed.

A Barsotti–Tate group is an important concept in the area of algebraic geometry and representation theory, particularly in the study of p-adic representations and finite field extensions. Named after mathematicians Francesco Barsotti and John Tate, these groups are essentially a kind of p-divisible group that has additional structure, allowing them to be classified and understood in terms of their representation theory.

In the context of algebraic groups and group theory, a **Borel subgroup** is a specific type of subgroup that is particularly important in the study of linear algebraic groups. Here are the key points regarding Borel subgroups: 1. **Definition**: A Borel subgroup of an algebraic group \( G \) is a maximal connected solvable subgroup of \( G \). This means that it cannot be contained in any larger connected solvable subgroup of \( G \).

Borel–de Siebenthal theory is a mathematical framework primarily associated with the study of compact Lie groups and their representations, particularly in the context of algebraic groups and symmetric spaces. The theory deals with the classification of maximal connected solvable subgroups, or Borel subgroups, in the context of semisimple Lie groups. It extends concepts of Borel subgroups from the language of algebraic groups to that of Lie groups.

Bruhat decomposition is a fundamental concept in the theory of Lie groups and algebraic groups, particularly in the study of algebraic varieties and symmetric spaces. It provides a way to decompose a group into pieces that can be analyzed more easily.

Cartier duality is a concept in the field of algebraic geometry and representation theory, particularly related to schemes and étale cohomology. It is named after the mathematician Pierre Cartier. At its core, Cartier duality establishes a relationship between a finite commutative group scheme over a field and its dual group scheme.

Chevalley's structure theorem is a fundamental result in the theory of algebraic groups and linear algebraic groups over algebraically closed fields. It provides a classification of connected algebraic groups over algebraically closed fields in terms of their semi-simple and unipotent parts.

Cohomological invariants are tools used in algebraic topology, algebraic geometry, and related fields to study the properties of topological spaces, algebraic varieties, or other mathematical structures through their cohomology groups. Cohomology provides a way to classify and distinguish topological spaces by associating algebraic invariants to them.

Complexification of a Lie group is a process that involves taking a real Lie group and extending it to a complex Lie group. This technique is useful in many areas of mathematics and theoretical physics because it allows for the application of complex analysis techniques to problems originally framed in the context of real manifolds.

A group is said to be diagonalizable if it can be represented in a certain way with respect to its action on a vector space, particularly in the context of linear algebra. More specifically, in the context of linear representations, a group is diagonalizable when its representation can be expressed in a diagonal form. In this context, consider a group \( G \) acting on a vector space \( V \) over some field, typically the complex numbers.

The Dieudonné module is an important concept in the field of arithmetic geometry, particularly in the study of the formal geometry over fields of positive characteristic, like finite fields. It arises within the context of formal schemes and is closely tied to the theory of p-divisible groups and formal groups.

Differential algebraic groups are mathematical structures that arise in the study of algebraic groups and differential equations. They combine concepts from algebraic geometry and differential geometry, specifically the theory of algebraic groups over differential fields. Here’s a more detailed breakdown of the concept: ### Algebraic Groups An algebraic group is a group that is also an algebraic variety, meaning it can be defined by polynomial equations. The group operations (multiplication and inversion) are also given by regular (i.

In mathematics, E6 refers to a specific complex Lie group, which is part of a classification of simple Lie groups. The E6 group is one of the five exceptional simple Lie groups, and it has applications in various fields, including theoretical physics, particularly in string theory and particle physics. The E6 group is often represented in terms of its root system, which consists of 72 roots in an 8-dimensional vector space.

In mathematics, "E7" typically refers to one of the exceptional Lie groups, which are important in various fields, including algebra, geometry, and theoretical physics. Specifically, E7 is a complex, simple Lie group of rank 7 that can be understood in terms of its root system and algebraic structure.

In mathematics, "F4" can refer to different concepts depending on the context. Here are a couple of potential interpretations: 1. **F_4 (Lie Algebra)**: In the context of Lie algebras, \( \mathfrak{f}_4 \) is one of the five exceptional simple Lie algebras.

In the context of group theory, a fixed-point subgroup refers to the set of elements in a group that remain unchanged under the action of a particular element or a group of elements, typically in the context of a group acting on a set. It's related to the idea of certain symmetries or invariances in that action. More formally, consider a group \( G \) acting on a set \( X \).

In mathematics, "G2" can refer to several concepts depending on the context. Here are a couple of prominent interpretations: 1. **Lie Group G2**: In the context of algebraic and geometric structures, G2 is one of the five exceptional simple Lie groups. It has a dimension of 14 and is associated with a specific type of symmetry.

The Generalized Jacobian is a mathematical concept that extends the idea of the Jacobian matrix, which is primarily used in calculus to describe how a function's output changes in response to small changes in its input. While the traditional Jacobian is applicable to smooth functions, the Generalized Jacobian is particularly useful in the context of nonsmooth analysis and optimization.

Geometric Invariant Theory (GIT) is a branch of algebraic geometry that studies the action of group actions on algebraic varieties, particularly focusing on understanding the properties of orbits and established notions of stability. It was developed primarily in the 1950s by mathematician David Mumford, building on ideas from group theory, algebraic geometry, and representation theory.

The Jacobson–Morozov theorem is a result in the representation theory of Lie algebras, specifically concerning the existence of certain embeddings of semisimple Lie algebras.

The Kazhdan–Margulis theorem is a result in the field of geometry and group theory, specifically concerning the behavior of discrete groups of isometries in the context of hyperbolic geometry. It was formulated by mathematicians David Kazhdan and Gregory Margulis in the 1970s. The theorem primarily addresses the structure of lattices in semi-simple Lie groups, particularly focusing on the behavior of certain types of actions of these groups on homogeneous spaces.

The Kempf vanishing theorem is a result in algebraic geometry that deals with the behavior of sections of certain vector bundles on algebraic varieties, particularly in the context of ample line bundles. Named after G. R. Kempf, the theorem addresses the vanishing of global sections of certain sheaves associated with a variety.

The Kneser–Tits conjecture is a statement in the field of algebraic groups and the theory of group actions, particularly concerning the structure of algebraic groups and their associated buildings. It was proposed by mathematicians Max Kneser and Jacques Tits. The conjecture pertains to the relationship between a certain class of algebraic groups defined over a field and their maximal compact subgroups.

Kostant polynomials are a class of polynomials that arise in the study of Lie algebras, representation theory, and several areas of algebraic geometry. They were introduced by Bertram Kostant in his work on the structure of semisimple Lie algebras and their representations. In particular, Kostant polynomials are closely associated with the weights of representations of a Lie algebra and its root system.

Lang's theorem is a result in the field of algebraic geometry, specifically related to the properties of algebraic curves. It is named after the mathematician Serge Lang. The theorem primarily concerns algebraic curves and their points over various fields, specifically in the context of rational points and rational functions. One important version of Lang's theorem states that a smooth projective curve over a number field has only finitely many rational points unless the curve is of genus zero.

Langlands decomposition is a concept in the context of representation theory of Lie groups, specifically related to the structure of semisimple Lie algebras and their representations.

Tits indices, named after the mathematician Jacques Tits, are a concept in the area of group theory and algebraic groups, particularly in the study of algebraic group representations and the structure of certain algebraic objects. Irreducible Tits indices are used to classify the irreducible representations of a group in relation to the structure of the group and its associated algebraic objects.

The Mumford–Tate group is a concept from algebraic geometry and number theory that arises in the study of abelian varieties and the associated Hodge structures. It is named after mathematicians David Mumford and John Tate. In the context of algebraic geometry, an abelian variety is a projective algebraic variety that has a group structure.

In the context of algebraic groups and representation theory, a pseudo-reductive group is a certain type of algebraic group that generalizes the notion of reductive groups. While reductive groups are well-studied and have nice properties, pseudo-reductive groups allow for a more general framework that still retains many desirable features.

In the context of algebraic groups, the **radical** refers to a specific type of subgroup that is closely related to the structure of the group itself. More formally, the radical of an algebraic group \( G \) is defined as the largest normal solvable subgroup of \( G \). ### Key Concepts: 1. **Algebraic Group**: An algebraic group is a group that is also an algebraic variety, meaning it can be defined by polynomial equations.

A **restricted Lie algebra** is a special type of Lie algebra that comes equipped with a unary operation called the "p-th power" which generalizes the notion of taking powers of elements in the context of Lie algebras. This concept is particularly important in the study of Lie algebras over fields of characteristic \( p \), where \( p \) is a prime number.

In the context of algebraic groups and Lie algebras, a **root datum** is a structured way of encoding certain aspects of the symmetries and properties of these mathematical objects. Specifically, a root datum consists of the following components: 1. **A finite set of roots**: These are usually vectors in a Euclidean space, which can be thought of as directions that reflect the symmetries of the system.

The Rost invariant is an important concept in the field of algebraic groups, particularly in relation to the study of quadratic forms and the theory of \(K\)-theory. It is named after the mathematician Ulrich Rost. The Rost invariant is defined in the context of central simple algebras, specifically those over a field. More concretely, it deals with the classification of certain types of quadratic forms and their behavior under various operations.

In mathematics, particularly in the area of algebraic geometry and number theory, a Serre group generally refers to a certain type of group that is associated with the work of Jean-Pierre Serre, a prominent French mathematician. There are different contexts in which "Serre group" may be used, but one of the more common references involves the concept related to *Serre's conjectures* in the theory of abelian varieties and algebraic groups.

A Severi–Brauer variety, named after the mathematicians Francesco Severi and Hans von Brauer, is a specific type of algebraic variety that is related to the study of division algebras and central simple algebras in algebraic geometry.

In the context of the theory of algebraic groups, particularly in the study of the general linear group \( GL(n, \mathbb{C}) \) or similar groups over other fields, a **Siegel parabolic subgroup** is a particular type of parabolic subgroup that is associated with a certain block upper triangular structure.

A Spaltenstein variety is a specific type of algebraic variety that is studied in the context of representation theory and algebraic geometry, particularly in relation to the study of finite dimensional representations of algebraic groups or algebraic varieties. Spaltenstein varieties arise in the context of the so-called "nilpotent cones." More specifically, they can be associated with certain types of objects called "nilpotent elements" in the representation theory of Lie algebras or algebraic groups.

The Tamagawa number is a concept in the field of number theory, specifically in the study of algebraic groups and arithmetic geometry. It is associated with a connected reductive algebraic group defined over a global field, such as a number field or a function field.

The Taniyama group, named after mathematician Yutaka Taniyama, is a group in the context of number theory that is closely related to the study of elliptic curves and modular forms. It is particularly famous for its connection to the Taniyama-Shimura-Weil conjecture, which posited that every elliptic curve over the rational numbers is associated with a modular form.

A **Torus action** refers to the action of a torus (typically a compact, connected Lie group isomorphic to the product of several circles, denoted as \(T^n\), where \(n\) is the number of circles) on some space, often a manifold. This concept arises in various areas of mathematics, including differential geometry, algebraic geometry, and symplectic geometry.

In the context of representation theory, the "trace field" of a representation typically refers to the field over which the representations of a group or algebra are defined, particularly when considering the trace of endomorphisms associated with the representation.

The Verschiebung operator, also known as the shift operator, is a mathematical operator used in various fields, including quantum mechanics and functional analysis. The term "Verschiebung" is German for "shift," and the operator is typically denoted by \( S \). In the context of quantum mechanics, for example, the shift operator can shift states in a Hilbert space.

Weil's conjecture on Tamagawa numbers is a part of the broader framework concerning algebraic groups and number theory, and specifically relates to the study of algebraic groups over global fields (like number fields or function fields). The conjecture connects the structure of algebraic groups to certain arithmetic invariants known as Tamagawa numbers.

Weyl modules are a family of representations associated with Lie algebras and are particularly important in the representation theory of semisimple Lie algebras. They are named after Hermann Weyl, who made significant contributions to the field of representation theory. ### Definition For a semisimple Lie algebra \(\mathfrak{g}\) over a field, a Weyl module \(V_\lambda\) is constructed for a given dominant integral weight \(\lambda\).

An étale group scheme is a concept from algebraic geometry and the theory of group schemes. It can be understood in the context of scheme theory, which is a branch of mathematics that deals with geometric objects defined by polynomial equations, among other things. ### Group Schemes First, let's break down the term "group scheme." A group scheme is a scheme equipped with a group structure.

Combinatorial group theory is a branch of mathematics that studies groups by using combinatorial methods and techniques. It focuses on understanding the properties of groups through their presentations, generators, and relations. The main goal is to analyze and classify groups by examining how these elements can be combined and related in various ways.

The concept of an "absolute presentation" of a group is a more advanced topic in group theory, especially in algebraic topology and geometric group theory. It provides a way to describe groups using generators and relations in a way that is independent of the specific context or properties associated with the group.

The automorphism group of a free group is a fundamental object in group theory and algebraic topology. Let \( F_n \) denote a free group on \( n \) generators. The automorphism group of \( F_n \), denoted as \( \text{Aut}(F_n) \), consists of all isomorphisms from \( F_n \) to itself. This group captures the symmetries of the free group.

The Baumslag–Solitar groups are a class of finitely presented groups, introduced by the mathematicians Gilbert Baumslag and Donald Solitar. They are significant in the study of group theory and have interesting properties related to their structure and actions.

The term "commutator collecting process" isn't a standard phrase in mainstream disciplines, so it might refer to specific contexts or fields, like physics, mathematics, or possibly even a particular area of study within abstract algebra or quantum mechanics. In quantum mechanics, a "commutator" refers to an operator that measures the extent to which two observables fail to commute (i.e., the extent to which the order of operations matters).

The "Freiheitssatz," or "freedom theorem," is a concept in mathematical logic and model theory, particularly in the context of formal languages.

In the context of topology and geometry, a **fundamental polygon** is a concept used to describe a polyhedral representation of a surface, particularly in the study of covering spaces and orbifolds. Here's a breakdown of the idea: 1. **Basic Definition**: A fundamental polygon is a two-dimensional polygon that serves as a model for the surface of interest. It provides a way to visualize and analyze the properties of that surface.

The Hall–Petresco identity is a mathematical result in the field of complex analysis, specifically related to the study of analytic functions and power series. It describes a relationship involving the coefficients of power series in connection with holomorphic functions defined in a disk.

The Herzog–Schönheim conjecture is a conjecture in the field of algebraic geometry and commutative algebra. It concerns the properties of ideals in polynomial rings or local rings. Specifically, it relates to the asymptotic behavior of the growth of the lengths of certain graded components of ideals.

In group theory, the concept of "normal form" can refer to a variety of representations that provide a canonical way to express elements in certain types of groups, particularly free groups and free products of groups. ### Normal Form for Free Groups A **free group** is a group where the elements can be represented as reduced words over a set of generators, with no relations other than those that are necessary to satisfy the group axioms (e.g., inverses for each generator).

The concept of an SQ-universal group arises in the context of group theory and, more generally, plays a role in the study of model theory and the interplay between algebra and logic. An **SQ-universal group** is a type of group that satisfies certain properties with respect to a specific class of groups known as **SQ** (stable, quotient) groups. The term "universal" indicates that this group can realize all finite SQ-types over the empty set.

Tietze transformations are a method in topology used to extend a continuous function defined on a subspace of a topological space to the whole space.

The Von Neumann conjecture is a mathematical conjecture related to the field of game theory and the concept of strategic behavior in games. More specifically, it is concerned with the optimal strategies in two-player games and provides insights into the nature of equilibria in these types of games.

Coxeter groups are abstract algebraic structures that arise in various areas of mathematics, including geometry, group theory, and combinatorics. They are defined by a particular type of presentation that involves reflections across hyperplanes in Euclidean space, but they can also be studied in a more abstract way.

Bruhat order is a partial order on the elements of a Coxeter group, particularly related to the symmetric group and general linear groups. It provides a way to compare the "sizes" or "positions" of elements based on their factorizations into simple reflections.

In mathematics, particularly in the study of reflection groups and Coxeter groups, a **Coxeter element** is a specific type of element that is associated with the generating reflections of a Coxeter group. More formally, a Coxeter group is defined by a set of generators that satisfy certain relations, typically corresponding to reflections across hyperplanes in a geometric space. A Coxeter element is typically constructed by taking a set of generators of the Coxeter group and forming their product in a specific order.

A Coxeter group is a special type of group that can be defined geometrically using reflections in Euclidean spaces. These groups are named after H.S.M. Coxeter, who studied their properties and relationships to various geometrical structures. ### Basic Definition: A Coxeter group is defined by a set of generators subjected to specific relations. These relations are based on the angles between the reflections corresponding to the generators.

A Coxeter–Dynkin diagram is a graphical representation used to describe finite and infinite reflection groups, which are important in various areas of mathematics, including geometry, algebra, and theoretical physics. These diagrams are named after mathematicians Harold Scott MacDonald Coxeter and Jacques Dynkin. ### Key Features of Coxeter–Dynkin Diagrams: 1. **Vertices**: Each vertex of the diagram represents a simple root of a root system associated with a given reflection group.

In the context of Coxeter groups, the **longest element** refers to a particular element of the group that can be identified based on its maximal length with respect to the generating set specified by the Coxeter diagram. A **Coxeter group** is defined by a set of generators and relations that can be represented by a diagram (called a Coxeter diagram) where each generator corresponds to a vertex.

A reflection group is a mathematical concept in the field of group theory, specifically in the study of symmetry. It is a type of group that consists of reflections across hyperplanes in a given vector space. Reflection groups can be thought of as the symmetries of geometric objects that can be achieved through reflections. ### Definitions and Properties: 1. **Reflections**: A reflection in a vector space is a linear transformation that flips points across a hyperplane.

Functional subgroups are specific categories or subdivisions within a larger organization or system that focus on a particular function or area of expertise. These subgroups are typically formed to enhance efficiency, streamline processes, and improve specialization in various tasks and responsibilities. For example, in a corporate setting, functional subgroups could include: 1. **Human Resources** - Focused on recruitment, employee relations, training, and benefits management.

In group theory, a **fitting subgroup** is a concept related to the structure of finite groups. Specifically, the Fitting subgroup of a group \( G \), denoted as \( F(G) \), is defined as the largest nilpotent normal subgroup of \( G \). ### Key Points about Fitting Subgroup: 1. **Nilpotent Group**: A group is nilpotent if its upper central series terminates in the whole group after finitely many steps.

In the context of group theory, an automorphism is an isomorphism from a group to itself. More formally, let \( G \) be a group. An automorphism is a function \( \phi: G \to G \) that satisfies the following properties: 1. **Homomorphism**: For all elements \( a, b \in G \), \( \phi(ab) = \phi(a) \phi(b) \).

In mathematics, specifically in the field of group theory and abstract algebra, an automorphism group is a concept that involves the symmetries of a mathematical structure. ### Definition An **automorphism** is an isomorphism from a mathematical object to itself. In other words, it is a bijective mapping that preserves the structure of the object.

In the context of mathematics, particularly in group theory and the study of algebraic structures, the term "quotientable automorphism" is not a standard terminology widely recognized in classic mathematical literature. However, I can help clarify two concepts that might relate to this phrase: 1. **Automorphism**: An automorphism is a structure-preserving map from a mathematical object to itself that has an inverse.

"Group products" can refer to a couple of different concepts depending on the context, but it generally relates to a category of goods that are grouped together based on certain characteristics. Here are a few interpretations: 1. **Marketing and Retail**: In marketing, group products can refer to items that are sold together, often because they complement each other.

In group theory, the **direct product** (also known as the **Cartesian product** or **product group**) of two groups combines the two groups into a new group. Here's a detailed explanation: ### Definition: Let \( G \) and \( H \) be two groups.

In group theory, the "product of group subsets" typically refers to the operation of combining elements from two subsets of a group to form new elements, often resulting in another subset within the group.

The semidirect product is a construction in group theory that allows you to combine two groups in a specific way, providing a means to build new groups from known ones. It is particularly useful in the study of groups with a certain structure and in applications in various areas of mathematics, including geometry and physics.

Infinite group theory typically refers to the study of groups that are infinite in size, which can include a wide variety of mathematical structures in the field of abstract algebra. In mathematics, a group is a set equipped with an operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverses.

In group theory, the concept of commensurability relates to the way in which two groups can be compared based on their subgroups and certain structural properties. Two groups \( G \) and \( H \) are said to be **commensurable** if they share a common finite-index subgroup.

An **elementary amenable group** is a type of group in the field of group theory, specifically within the area of descriptive set theory and ergodic theory. Groups are classified as elementary amenable if they can be constructed from finite groups through a combination of certain operations.

An **FC-group** (or **Finite Class Group**) is a specific type of group in the field of group theory, a branch of mathematics. FC-groups are characterized by the property that every element has a finite number of conjugates, meaning that the set of conjugates for each element is finite.

A *free-by-cyclic group* is a specific type of group that can be thought of as a combination of two structures: a free group and a cyclic group. More formally, a free-by-cyclic group is a group \( G \) that can be expressed in the form: \[ G = F \rtimes C \] where \( F \) is a free group and \( C \) is a cyclic group.

Gromov's theorem on groups of polynomial growth states that any finitely generated group with polynomial growth is virtually nilpotent. This theorem is a significant result in geometric group theory and has important implications for the structure of groups.

Higman's embedding theorem is a result in the field of formal languages and automata theory, specifically relating to the study of recursively enumerable languages and context-free languages. The theorem provides a way to understand the structure of certain algebraic objects associated with these languages.

A Hopfian group is a type of group that satisfies a specific property related to its endomorphisms. Specifically, a group \( G \) is called a Hopfian group if every surjective (onto) endomorphism \( f: G \to G \) is an isomorphism.

The infinite dihedral group, usually denoted as \( D_{\infty} \) or sometimes \( D_{\infty}^* \), is a mathematical structure in group theory that extends the concept of the dihedral groups. While the finite dihedral group \( D_n \) represents the symmetries of a regular polygon with \( n \) sides (including rotations and reflections), the infinite dihedral group captures symmetries of an infinite linear arrangement.

In group theory, a branch of abstract algebra, an **infinite group** is a group that contains an infinite number of elements. In other words, if the cardinality (size) of the group is not a finite number, then the group is classified as infinite. Infinite groups can be categorized into various types based on their structure and properties.

A **locally finite group** is a type of group in the field of abstract algebra. Specifically, a group \( G \) is called locally finite if every finite subset of \( G \) generates a finite subgroup of \( G \). In other words, for any finite subset \( S \) of \( G \), the subgroup generated by \( S \), denoted by \( \langle S \rangle \), is finite.

A **Pro-p group** is a type of mathematical object in the field of group theory, particularly in the area of profinite groups. More specifically, a Pro-p group is a topological group that is both locally compact and totally disconnected, and it is defined as an inverse limit of finite groups whose orders are powers of a prime \( p \).

A **profinite group** is a type of topological group that has a very specific structure. These groups are characterized by several key features: 1. **Definition**: A profinite group is a compact, totally disconnected, Hausdorff topological group that is isomorphic to the inverse limit of a system of finite groups. In more intuitive terms, you can think of profinite groups as "limits" of finite groups that retain a group structure.

The Prüfer rank, also known as the Prüfer order, is a concept from the field of algebraic topology and algebraic K-theory that applies to modules, particularly in relation to Prüfer domains. It is a measure of the "size" of a module, similar to the rank of a vector space, but adapted for module theory.

A group \( G \) is called **residually finite** if for every nontrivial element \( g \in G \) (i.e., \( g \neq e \), where \( e \) is the identity element of the group), there exists a finite group \( H \) and a group homomorphism \( \varphi: G \to H \) such that \( \varphi(g) \neq \varphi(e) \).

The residue-class-wise affine group is a mathematical concept that arises in the context of group theory, specifically in relation to affine transformations and modular arithmetic. To understand it better, let's break down the terms involved: 1. **Affine Transformation**: An affine transformation can be viewed as a function that maps points from one vector space to another while preserving points, straight lines, and planes.

A Tarski monster group is a specific type of mathematical group that has some intriguing properties, particularly in the field of group theory. Specifically, a Tarski monster group is an infinite group in which every non-trivial subgroup has a prime order \( p \).

Thompson groups are a family of groups that arise in the area of geometric group theory, named after the mathematician J. G. Thompson who introduced them. They are defined in the context of homeomorphisms of the unit interval \([0, 1]\) and can be understood as groups of piecewise linear homeomorphisms.

Tits' alternative is a concept from group theory, named after mathematician Jacques Tits. It refers to a criterion for determining whether a given group is either "a linear group" or "a free group." More formally, it involves the classification of certain types of groups based on their actions on vector spaces.

The term "verbal subgroup" can refer to different concepts depending on the context, so it’s important to clarify where you might be encountering it. However, in mathematical group theory, a verbal subgroup typically refers to a subgroup generated by certain types of elements of a group that satisfy specific equations or properties.

The term "Z-group" can refer to different concepts depending on the context. Here are a few interpretations: 1. **Mathematics**: In group theory, a Z-group could refer to a group that is isomorphic to the additive group of integers, denoted as \( \mathbb{Z} \). This is a fundamental example of an infinite cyclic group.

Moonshine theory, also known simply as "moonshine," is a fascinating area of research in mathematics that explores deep connections between number theory, algebra, and mathematical physics. The term originally arises from the surprising mathematical phenomena discovered by John McKay in 1978 and further developed by others, including Richard Borcherds and Hollis Lang. At its core, moonshine refers to the conjectural relationships between finite groups and modular forms.

Kac–Moody algebras are a class of infinite-dimensional Lie algebras that generalize the concept of finite-dimensional semisimple Lie algebras. They are named after Victor G. Kac, who introduced them in the 1960s as a way to study certain symmetries in mathematical physics and representation theory. A Kac–Moody algebra is defined by a generalized Cartan matrix, which captures the relationships between the root system of the algebra.

The Monster Lie algebra is associated with the Monster group, which is the largest of the sporadic simple groups in group theory. The Monster group itself has fascinating connections to various areas of mathematics, including group theory, number theory, and algebraic geometry. The Monster Lie algebra can be thought of as an infinite-dimensional Lie algebra that arises in the study of the Monster group. It is defined by a set of generators and relations that reflect the symmetries and structural properties of the Monster group.

The Monster group, denoted as \( \mathbb{M} \) or sometimes \( \text{Mon} \), is the largest of the 26 sporadic simple groups in group theory, a branch of mathematics that studies algebraic structures known as groups. It was first discovered by Robert Griess in 1982 and has a rich structure that connects various areas of mathematics, including number theory, geometry, and mathematical physics.

Supersingular primes are an important concept in the context of moonshine theory, which is a branch of number theory that connects two seemingly disparate areas: modular forms and finite group theory. More specifically, moonshine theory is famous for exploring the relationship between certain mathematical structures—the Monster group, the largest of the so-called sporadic simple groups, and modular functions.

In mathematics, specifically in group theory, an **ordered group** is a group that is equipped with a total order that is compatible with the group operation. This means that not only do the group elements have a way to be compared (one element can be said to be less than, equal to, or greater than another), but the group operation respects this order.

An Archimedean group is an important concept in the field of mathematics, particularly within the context of ordered groups. An ordered group is a group that is equipped with a total order that is compatible with the group operation.

A cyclically ordered group is a mathematical structure that extends the concept of a group by introducing a specific type of total order compatible with the group operation. More formally, a cyclically ordered group is a group \( G \) equipped with a binary relation \( < \) that satisfies certain conditions to ensure that the order is "cyclic.

The Hahn embedding theorem is a result in functional analysis, particularly in the study of ordered vector spaces and topological vector spaces. It is named after the mathematician Hans Hahn. The theorem states that every ordered vector space can be embedded into a space of real-valued functions.

A **linearly ordered group** is a mathematical structure that combines the properties of a group with those of a linear order. More specifically, it is a group \( G \) equipped with a total order \( < \) that is compatible with the group operation.

An ordered field is a field \( F \) equipped with a total order \( \leq \) that is compatible with the field operations. This means that the order satisfies the following properties: 1. **Totality**: For any two elements \( a, b \in F \), one of the following holds: \( a \leq b \) or \( b \leq a \).

An **ordered ring** is a mathematical structure that combines the properties of a ring with a total order. More formally, an ordered ring is defined as a ring \( R \) together with a total order \( \leq \) that satisfies certain compatibility conditions with the ring operations (addition and multiplication).

A **partially ordered group** (POG) is an algebraic structure that combines the concepts of a group and a partial order. Formally, a group \( G \) is equipped with a binary operation (usually denoted as multiplication or addition) and satisfies the group properties—closure, associativity, existence of an identity element, and existence of inverses.

A **Riesz space** (also known as a **vector lattice**) is a specific type of ordered vector space that combines both vector space and lattice structures.

P-groups, or *p-groups*, are a specific type of group in the field of abstract algebra, particularly in the study of group theory. A group \( G \) is classified as a p-group if the order (the number of elements) of the group is a power of a prime number \( p \). Formally, this can be expressed as: \[ |G| = p^n \] for some non-negative integer \( n \).

Coclass, also known as co-class or co-classification, is a mathematical concept primarily used in the field of group theory, more specifically in the study of finite groups and their subgroups. It refers to a particular classification of groups based on shared properties of their Sylow subgroups.

The term "Extra Special Group" is not widely defined in common literature, organizations, or terminology as of my last knowledge update in October 2021. It could refer to a specific organization, initiative, or group focusing on unique or niche areas, but without additional context, it's challenging to provide an accurate description.

The Focal Subgroup Theorem is a concept in the area of algebraic topology and group theory, particularly relating to finite group actions and their relationships to fixed point sets in topological spaces. In more detail, the Focal Subgroup Theorem often pertains to the study of groups acting on topological spaces and examines the interaction between the group action and the topology of the space.

In group theory, which is a branch of abstract algebra, a **P-group** is a type of group that plays an important role in the study of finite groups. Specifically, a P-group is defined as a group in which the order (the number of elements) of every element is a power of a prime number \( p \).

A **powerful \( p \)-group** is a special type of \( p \)-group (a group where the order of every element is a power of a prime \( p \)) that satisfies certain conditions regarding its commutator structure.

A **regular p-group** is a specific type of finite group that is defined in the context of group theory, particularly in relation to \( p \)-groups. A **\( p \)-group** is a group where the order (the number of elements) of the group is a power of a prime number \( p \).

In the context of finite group theory, a "special group" typically refers to a type of group that has specific properties. One common usage is related to **special linear groups**. However, the term could also refer to **special groups in the context of group extensions** or other specific constructions in group theory.

The Sylow theorems are a set of results in group theory, a branch of abstract algebra. They provide important information about the subgroups of a finite group, particularly regarding the existence and properties of p-subgroups, where p is a prime number.

Representation theory of groups is a branch of mathematics that studies how groups can be represented through linear transformations of vector spaces. More formally, a representation of a group \( G \) is a homomorphism from \( G \) to the general linear group \( GL(V) \) of a vector space \( V \). This means that each element of the group is associated with a linear transformation, preserving the group structure.

Representation theory of Lie groups is a branch of mathematics that studies how Lie groups can be represented as groups of transformations on vector spaces. More formally, a representation of a Lie group \( G \) is a homomorphism from \( G \) to the general linear group GL(V) of invertible linear transformations on a vector space \( V \). This allows one to study properties of the group \( G \) through linear algebra and the geometry of vector spaces.

Representation theory of finite groups is a branch of mathematics that studies how groups, particularly finite groups, can be represented through linear transformations of vector spaces. In simpler terms, it examines how abstract groups can be manifested as matrices or linear operators acting on vector spaces.

Unitary representation theory is a branch of mathematics and physics that studies how groups can be represented through unitary operators on Hilbert spaces. In this context, a **unitary representation** of a group \( G \) is a homomorphism from the group \( G \) into the group of unitary operators on a Hilbert space \( H \).

The "Atlas of Lie Groups and Representations" is a comprehensive project that provides a detailed database of information about Lie groups and their representations. Lie groups are mathematical structures that are used to describe continuous symmetries, and they play a pivotal role in many areas of mathematics and theoretical physics, particularly in the study of differential equations, geometry, and quantum mechanics.

B-admissible representation is a concept in the realm of representation theory, particularly in the study of p-adic groups and their representations. The notion arises in the context of understanding how representations of a given group can be analyzed through the properties of certain subgroups. In more formal terms, let \( G \) be a p-adic group, and let \( B \) be a Borel subgroup of \( G \).

Character theory is a branch of mathematics, specifically within the field of representation theory of finite groups and algebra. It studies the characters of group representations, which are complex-valued functions that provide insight into the structure of the group. In essence, a character of a group representation is a function that assigns to each group element a complex number, which is the trace of the corresponding linear transformation in a representation.

In mathematics and physics, particularly in the context of complex numbers and quantum mechanics, the term "complex conjugate representation" can have specific meanings depending on the context.

Complex representation refers to the method of expressing mathematical or physical concepts using complex numbers, which are numbers that have both a real part and an imaginary part.

The concept of corepresentations of unitary and antiunitary groups arises primarily in the context of representation theory, which studies how groups act on vector spaces through linear transformations. In quantum mechanics and in many areas of physics, these groups often illustrate symmetries of systems, where unitary and antiunitary operators play significant roles. ### Unitary Groups Unitary operators are linear operators associated with a unitary group, which is a group of transformations that preserve inner products in complex vector spaces.

Dual representation refers to the ability to understand and represent the same information in different ways or formats. This concept is often discussed in various fields, including psychology, education, and cognitive science, particularly in relation to learning and comprehension. In the context of cognitive development, particularly in children, dual representation is exemplified by the ability to understand that a model or symbol (such as a map or a scale model) can represent something else in the real world.

Fontaine's period rings are a concept in the field of arithmetic geometry and number theory, specifically related to p-adic Hodge theory. They were introduced by Pierre Fontaine in the context of understanding the relationships between different types of cohomology theories, particularly for p-adic representations of the absolute Galois group of a p-adic field. More concretely, Fontaine's period rings provide a framework for studying p-adic Galois representations and their associated periods.

The Frobenius–Schur indicator is a concept from representation theory, particularly concerning finite groups and their representations. It provides a way to classify irreducible representations of a finite group with respect to their behavior under certain types of symmetry. In more specific terms, the Frobenius–Schur indicator is defined for an irreducible representation of a finite group \( G \) over a field \( K \) (typically, the complex numbers).

In the field of harmonic analysis and representation theory, a **Gelfand pair** is a specific type of mathematical structure that arises when studying the representations of groups. More concretely, a Gelfand pair consists of a pair of groups (typically a group \( G \) and a subgroup \( H \)) such that the algebra of \( H \)-invariant functions on \( G \) is particularly "nice" for some representation theory considerations.

The Gelfand–Raikov theorem is a result in functional analysis and, more specifically, in the theory of Hilbert spaces. It provides conditions under which a certain type of operator can be approximated by a sequence of rank-one operators.

A **group ring** is a mathematical structure that is used in abstract algebra, combining concepts from both group theory and ring theory. More specifically, if \( G \) is a group and \( R \) is a ring, the group ring \( R[G] \) is a new ring constructed from these two objects. ### Construction of the Group Ring 1.

In mathematics, particularly in the context of algebra and representation theory, the term "K-finite" usually refers to elements in a representation (or module) of a group or algebra that have a certain finiteness property related to a subgroup \( K \). For example, in the representation theory of Lie groups, a representation is said to be K-finite if every vector in the representation space can be approximated by finite sums of vectors transformed by elements of a compact subgroup \( K \).

In the context of linear algebra and matrix theory, the term "matrix coefficient" can refer to a few different concepts depending on the specific area of study. Here are some possible interpretations: 1. **Matrix Elements**: In a square matrix, each entry or element is often referred to as a coefficient.

The McKay conjecture is a hypothesis in the field of representation theory and algebraic geometry, particularly regarding the relationship between finite groups and certain geometric structures. Formulated by John McKay in the 1980s, the conjecture specifically connects the representation theory of finite groups (especially simple groups) and the geometry of algebraic varieties.

Molien's formula is a result in invariant theory that provides a way to calculate the generating function of the dimensions of the spaces of invariants of polynomial functions under the action of a group. Specifically, it can be used to find the generating function for the dimensions of the invariant polynomials under the action of a linear group.

The Multiplicity-One Theorem is a concept in the field of algebraic geometry, particularly in the study of algebraic varieties and their singularities. It is often applied in the context of intersections of algebraic varieties, particularly in relation to issues involving the dimension and the multiplicity of points of intersection. In general terms, the Multiplicity-One Theorem states that if two varieties intersect transversely at a point, then the intersection at that point has multiplicity one.

P-adic Hodge theory is a branch of mathematics that lies at the intersection of algebraic geometry, number theory, and representation theory. It provides a framework for understanding the behavior of p-adic forms and their connections to classical geometry.

Partial group algebra is a mathematical structure that arises in the context of representation theory and algebra. It is related to the study of groups and their actions, particularly in situations where you want to consider a group acting on a set but only on a portion of that set.

In the context of group theory, the regular representation of a group provides a way to represent group elements as linear transformations on a vector space.

In the context of representation theory and algebra, a **representation rigid group** generally refers to a group for which the representations exhibit a certain rigidity or inflexibility. The term can be more specific in certain contexts or research areas but is often associated with groups whose representations are highly structured.

Representation theory of diffeomorphism groups is a mathematical framework that studies the actions of diffeomorphism groups on various spaces, particularly in the context of differential geometry, dynamical systems, and mathematical physics. Diffeomorphism groups are groups consisting of all smooth bijective mappings (diffeomorphisms) from a manifold to itself, equipped with a smooth structure, and they play a crucial role in understanding the symmetries and geometric structures of manifolds.

The Schur orthogonality relations are a set of mathematical statements that arise in the context of representation theory, particularly concerning the representations of the symmetric group and the general linear group. These relations provide a way to understand how different irreducible representations (irreps) of a group are related to one another through their characters.

Schur–Weyl duality is a fundamental result in representation theory that describes a deep relationship between two types of algebraic structures: the symmetric groups and the general linear groups. Specifically, it provides a duality between representations of the symmetric group \( S_n \) and representations of the general linear group \( GL(V) \) (where \( V \) is a finite-dimensional vector space) for a fixed \( n \).

Springer correspondence is a concept in the context of representation theory of Lie algebras, particularly associated with the theory of vertex operator algebras and the study of affine Lie algebras. The correspondence refers to a deep and intricate relationship between certain types of representations of vertex operator algebras and representations of affine Lie algebras.

Tempered representations are a concept from the field of representation theory, particularly in the context of reductive groups over local fields. They are an important part of the harmonic analysis on groups and play a vital role in the study of automorphic forms and number theory. In more detail: 1. **Context**: Tempered representations arise in the study of the representations of reductive groups over a local field (like the p-adic numbers or the real numbers).

Subgroup properties in group theory refer to certain characteristics or conditions that a subgroup of a given group may satisfy. These properties help in categorizing subgroups and understanding their structure relative to the larger group.

In group theory, an **abnormal subgroup** is a specific type of subgroup that captures certain properties related to the structure of the group. A subgroup \( H \) of a group \( G \) is called **abnormal** if it satisfies the following condition: For every \( g \in G \), if \( gH \) (the left coset of \( H \) in \( G \)) intersects with \( H \) non-trivially (i.e.

In group theory, a branch of abstract algebra, an **ascendant subgroup** of a group \( G \) is a specific type of subgroup that has a unique property concerning its relation to the whole group.

A **C-normal subgroup** is a concept from group theory, a branch of mathematics that studies the algebraic structures known as groups. A subgroup \( N \) of a group \( G \) is termed a **C-normal subgroup** if it satisfies certain conditions related to its normality.

In the context of group theory, a **Carter subgroup** is a specific type of subgroup associated with a finite group, particularly in the study of nilpotent and solvable groups. Specifically, a Carter subgroup is defined as follows: - It is a subgroup that is the intersection of all Sylow subgroups corresponding to its normalizer in the group.

In group theory, a branch of abstract algebra, a **central subgroup** refers to a subgroup that is contained in the center of a given group. The center of a group \( G \), denoted \( Z(G) \), is defined as the set of all elements \( z \in G \) such that \( zg = gz \) for all \( g \in G \). In other words, the center consists of all elements that commute with every other element in the group.

In group theory, a subgroup \( H \) of a group \( G \) is said to be **centrally closed** if it is closed under the operation of conjugation by elements of the center of \( G \).

A **characteristic subgroup** of a group \( G \) is a subgroup \( H \) that is invariant under all automorphisms of the group \( G \). This means that for any automorphism \( \phi \) of \( G \), the image \( \phi(H) \) is still a subgroup of \( G \) and is equal to \( H \) itself.

In group theory, a **conjugate-permutable subgroup** is a specific type of subgroup that has a particular property related to conjugation. A subgroup \( H \) of a group \( G \) is said to be conjugate-permutable if for every element \( g \in G \), the following condition holds: \[ H^g = gHg^{-1} \text{ satisfies } H^g \cap H \neq \emptyset.

In group theory, a **contranormal subgroup** is a type of subgroup with a particular relationship to normal subgroups and normality conditions in a larger group.

In group theory, a branch of abstract algebra, the term "descendant subgroup" refers to a subgroup that is generated by certain elements of a group and is contained within a larger structure, typically in the context of the subgroup lattice.

In the context of group theory, the concept of a **fully normalized subgroup** pertains to a subgroup that is maximal with respect to the property of being normal in a certain sense. Specifically, a subgroup \( H \) of a group \( G \) is said to be fully normalized if it is normal in every subgroup of \( G \) that contains it.

A Hall subgroup is a concept from group theory, specifically in the study of finite groups. It is named after Philip Hall, who introduced the concept in his work on groups and combinatorics.

A **malnormal subgroup** is a specific type of subgroup within group theory, particularly in the context of group actions and normal subgroups.

In group theory, a branch of abstract algebra, a **maximal subgroup** is a specific type of subgroup of a given group. A subgroup \( M \) of a group \( G \) is called a maximal subgroup if it is proper (meaning that it is not equal to \( G \)) and is not contained in any other proper subgroup of \( G \). In other words, there are no subgroups \( N \) such that \( M < N < G \).

In the context of group theory, particularly in the study of modular lattices and modular subgroups, a **modular subgroup** is a specific type of subgroup that satisfies the modular law.

A **normal subgroup** is a special type of subgroup in the context of group theory, which is a branch of abstract algebra. Let's define it more precisely. Given a group \( G \) and a subgroup \( N \) of \( G \): 1. **Subgroup**: A subgroup \( N \) must itself be a group under the operation defined on \( G \).

A **paranormal subgroup** is a concept in group theory, specifically in the area of finite group theory. A subgroup \( H \) of a group \( G \) is said to be paranormal if it meets a specific condition related to its normality and the structure of \( G \).

A **polynormal subgroup** is a concept from group theory, particularly in the study of group extensions and solvable groups. A subgroup \( N \) of a group \( G \) is called **polynormal** if for every finite sequence of subgroups \( H_1, H_2, \ldots, H_n \) of \( G \) such that: 1. \( H_1 \) is a subgroup of \( N \), 2.

A pronormal subgroup is a specific type of subgroup in group theory, particularly in the context of finite groups. A subgroup \( H \) of a group \( G \) is said to be **pronormal** if, for every \( g \in G \), the intersection of \( H \) with \( H^g \) (the conjugate of \( H \) by \( g \)) is a normal subgroup of \( H \).

A **quasinormal subgroup** is a concept in group theory, a branch of abstract algebra. A subgroup \( H \) of a group \( G \) is said to be quasinormal if it is permutable with every subgroup of \( G \).

In group theory, a **seminormal subgroup** is a particular type of subgroup within a group that is related to the concept of normality.

A semipermutable subgroup is a concept in the field of group theory, particularly in the study of group extensions and solvable groups. A subgroup \( H \) of a group \( G \) is called **semipermutable** if for every normal subgroup \( N \) of \( G \) such that \( N \) is a subset of \( H \), the subgroup \( H \) permutes with \( N \) in \( G \).

In group theory, a branch of abstract algebra, a **serial subgroup** refers to certain kinds of normal subgroups within a group. Specifically, a subgroup \( H \) of a group \( G \) is termed a serial subgroup if it can be expressed in a specific way in relation to the entire group \( G \) and to other subgroups.

In the context of group theory, a **special abelian subgroup** usually refers to a specific type of subgroup within a group, particularly in the theory of finite groups or in the study of Lie algebras.

In group theory, a **subnormal subgroup** is a specific type of subgroup that has a particular relationship with the larger group it is part of.

In group theory, a subgroup \( H \) of a group \( G \) is said to be **transitively normal** in \( G \) if it is normal in every subgroup of \( G \) that contains \( H \).

A **subgroup series** in group theory is a sequence of subgroups of a given group \( G \) that is organized such that each subgroup is a normal subgroup of the next one in the series.

In group theory, a branch of abstract algebra, a **central series** is a specific type of series of subgroups associated with a given group. It provides a way to study the structure of a group by breaking it down into simpler components.

The "Chief" series you are referring to might relate to a few different contexts, including literature, television, or a specific brand. However, one of the most notable references could be to the "Chief" series in the realm of literature or television featuring a character that holds a leadership position, such as Chief of Police or Chief Executive Officer, and often involves themes of crime, management, and problem-solving.

Fitting length, often used in contexts like pharmaceuticals, manufacturing, or engineering, refers to the length or dimension that is required for parts or components to fit together properly. In these contexts, achieving the correct fitting length is crucial for ensuring that components function as intended without issues such as misalignment, gaps, or mechanical failure.

Topological groups are a mathematical structure that combines concepts from both topology and group theory. Specifically, a topological group is a set equipped with two structures: a group structure and a topology, such that the group operations (multiplication and taking inverses) are continuous with respect to the topology.

Discrete groups are a type of mathematical structure studied primarily in the fields of abstract algebra and topology. Here's a breakdown of the concept: ### Definition A **discrete group** is a group \( G \) that is equipped with a discrete topology. In simpler terms, the group is a set of elements along with a binary operation (e.g.

An **amenable group** is a type of mathematical structure studied in the field of group theory, specifically in the study of topological groups and functional analysis. The concept of amenability is related to the ability of a group to have a certain type of "invariance" property under averaging processes. A group \( G \) is called **amenable** if it has a left-invariant mean.

Bohr compactification is a mathematical construction in the field of topological groups, particularly in the area of harmonic analysis and the theory of locally compact abelian groups. It is primarily associated with the study of the structure of such groups and their representations.

The Cantor cube, often denoted as \(2^\omega\) or \([0, 1]^\omega\), is a product space that arises in topology and set theory. It can be understood in a few different ways: 1. **Composition**: The Cantor cube is defined as the countable infinite product of the discrete space \(\{0, 1\}\).

Chabauty topology is a concept used in algebraic geometry and arithmetic geometry, specifically in the study of the spaces of subvarieties of algebraic varieties. It is named after the mathematician Claude Chabauty, who developed this topology in the context of algebraic varieties and their rational points. In the Chabauty topology, one can think about the space of closed subsets of a given topological space (often within a certain context such as algebraic varieties).

In the context of mathematics, specifically in the field of topology and group theory, a **compact group** is a group that is both compact as a topological space and a group in the sense of group operations. ### Definitions 1. **Topological Group**: A topological group is a set equipped with a group structure that is also a topological space, such that the group operations (multiplication and taking inverses) are continuous with respect to the topology of the space.

A **compactly generated group** is a type of topological group that can be characterized by the manner in which it is generated by compact subsets. Specifically, a topological group \( G \) is said to be compactly generated if there exists a compact subset \( K \subseteq G \) such that the whole group \( G \) can be expressed as the closure of the subgroup generated by \( K \).

A **continuous group action** is a mathematical concept that arises in the field of topology and group theory. Specifically, it involves a group acting on a topological space in a way that is compatible with the topological structure of that space. ### Definition: Let \( G \) be a topological group and \( X \) be a topological space.

In group theory, a branch of abstract algebra, a **covering group** is a concept that relates to the idea of covering spaces in topology, though it is used more specifically in the context of group representations and algebraic structures. A covering group can refer to a group that serves as a double cover of another group in the sense of group homomorphisms.

In the context of topological groups, the **direct sum** (often referred to as the **direct product**, especially in the category of groups) of a family of topological groups provides a way to combine these groups into a new topological group. The construction is analogous to that of the direct sum in vector spaces.

In the context of group theory, a *discontinuous group* usually refers to a group of transformations that is not continuous in a topological sense. This term can have different meanings depending on the mathematical context in which it is used, but here are two key interpretations: 1. **Mathematical Groups and Topology**: In general topology, a discontinuous group may refer to a group of homeomorphisms that do not form a continuous path between their elements.

In the context of topology and abstract algebra, an **extension** of a topological group refers to a way of constructing a new topological group from a known one by incorporating additional structure. This often involves creating a new group whose structure represents a combination of an existing group and a simpler group.

The Haar measure is an important concept in the area of harmonic analysis and abstract algebra, specifically in the context of topological groups. It is a way of defining a measure on a locally compact topological group that is left-invariant (or right-invariant), which means it remains unchanged (invariant) under the group's operations.

The Hilbert–Smith conjecture is a statement in the field of topology, particularly concerning group actions on topological spaces.

A homogeneous space is a mathematical structure that exhibits a high degree of symmetry. More formally, in the context of geometry and algebra, a homogeneous space can be defined as follows: 1. **Definition**: A space \(X\) is called a homogeneous space if for any two points \(x, y \in X\), there exists a symmetry operation (usually described by a group action) that maps \(x\) to \(y\).

The term "Identity component" can refer to different concepts depending on the context in which it is used. Here are a few interpretations across various fields: 1. **Mathematics**: In topology and algebra, the identity component of a topological space is the maximal connected subspace that contains the identity element. For a Lie group or a topological group, the identity component is the set of elements that can be path-connected to the identity element of the group.

Kazhdan's property (T) is a property of groups that was introduced by the mathematician David Kazhdan in the context of representation theory and geometric group theory. It is a strong form of compactness that relates to the representation theory of groups, particularly in how they act on Hilbert spaces.

Kronecker's theorem, also known as the Kronecker limit formula, is a result in number theory specifically related to the distribution of prime numbers and the behavior of certain algebraic objects. It can be particularly focused on the context of the theory of partitions or modular forms, but the term might refer to different results depending on the field.

A **locally compact group** is a type of topological group that has the property of local compactness in addition to the group structure. Let's break down the definitions: 1. **Topological Group**: A group \( G \) is equipped with a topology such that both the group operation (multiplication) and the inverse operation are continuous.

A **locally profinite group** is a type of group that is constructed from profinite groups, which are groups that are isomorphic to an inverse limit of finite groups. Formally, a locally profinite group can be defined as a group \( G \) that has a neighborhood basis at the identity consisting of open subgroups that are profinite.

A **loop group** is a concept from mathematics, particularly in the fields of algebraic geometry, differential geometry, and mathematical physics. It typically refers to a specific kind of group associated with loops in a manifold, particularly in the context of Lie groups.

In the context of Lie groups and algebraic groups, a **maximal compact subgroup** is a specific type of subgroup that has particular significance in the study of group structures. ### Definition: A **maximal compact subgroup** of a Lie group \( G \) is a compact subgroup \( K \) of \( G \) such that there is no other compact subgroup \( H \) of \( G \) that properly contains \( K \) (i.e.

A **monothetic group** is a term used in the context of taxonomy and systematics, particularly in the classification of organisms. It refers to a group of organisms that are united by a single common characteristic or a single attribute that defines that group. This characteristic is often a specific trait or combination of traits that all members of the group share, distinguishing them from organisms outside the group.

A **one-parameter group** is a mathematical concept primarily used in the fields of group theory and differential equations. It represents a continuous group of transformations that can be parametrized by a single real parameter, often denoted as \( t \).

A **paratopological group** is a mathematical structure that combines the concepts of group theory and topology, but with a relaxed condition on the topology. Specifically, a paratopological group is a set equipped with a group operation that is continuous in a weaker sense than standard topological groups.

The Peter–Weyl theorem is a fundamental result in the representation theory of compact topological groups. It describes how the regular representation of a compact group can be decomposed into irreducible representations. Here's a brief overview of the main points of the theorem: 1. **Compact Groups**: The theorem applies specifically to compact groups, which are groups that are also compact topological spaces. Examples include \(SU(n)\), \(SO(n)\), and \(U(n)\).

Positive real numbers are the set of numbers that are greater than zero and belong to the set of real numbers. This includes all the numbers on the number line to the right of zero, which can be represented as: - All whole numbers greater than zero (1, 2, 3, ...) - All fractions greater than zero (such as 1/2, 3/4, etc.) - All decimal numbers greater than zero (like 0.1, 2.

Quasiregular representation is a concept from the field of geometry and complex analysis, specifically within the study of quasiregular mappings. Quasiregular mappings are a generalization of holomorphic (complex analytic) functions, which allow for a broader class of functions including those that are not necessarily differentiable in the classical sense.

A "restricted product" typically refers to items that are subject to certain legal or regulatory limitations regarding their sale, distribution, or use. The specifics can vary widely depending on the context and jurisdiction, but here are some common categories of restricted products: 1. **Controlled Substances**: Pharmaceuticals or chemicals that are regulated due to their potential for abuse or harm (e.g., narcotics).

The Schwartz–Bruhat function, often simply referred to as the Schwartz function, is a type of smooth function that is rapidly decreasing. Specifically, it belongs to the space of smooth functions that decay faster than any polynomial as one approaches infinity. This type of function is especially important in various areas of analysis, particularly in the fields of distribution theory, Fourier analysis, and partial differential equations.

A **semitopological group** is a type of mathematical structure that combines aspects of group theory and topology. Specifically, it is a group \( G \) that is equipped with a topology such that the group operations, namely multiplication and taking inverses, satisfy specific continuity conditions, but not all of the usual requirements of a topological group.

A **topological group** is a mathematical structure that combines the concepts of a group and a topological space. Specifically, a topological group is a set equipped with two structures: a group structure and a topology that makes the group operations continuous.

A **topological ring** is a mathematical structure that combines the concepts of a ring and a topology. Specifically, a topological ring is a ring \( R \) that is also equipped with a topology such that the ring operations (addition and multiplication) are continuous with respect to that topology.

A totally disconnected group is a type of topological group in which the only connected subsets are the singletons, meaning that the only connected subsets of the group consist of individual points. This concept can be understood in the context of topological spaces and group theory. In more formal terms, a topological group \( G \) is said to be totally disconnected if for every two distinct points in \( G \), there exists a neighborhood around each point such that these neighborhoods do not intersect.

In various contexts, a (B, N) pair can refer to different concepts depending on the field of study.

An **acylindrically hyperbolic group** is a type of group in geometric group theory that generalizes the concept of hyperbolic groups. These groups are characterized by a specific type of action they have on a $\textit{proper geodesic metric space}$.

The affine group is a mathematical concept that arises in the context of geometry and linear algebra. It is essentially a group that consists of affine transformations, which are a generalization of linear transformations that include translations.

The concept of an "approximate group" arises in the field of group theory and is particularly relevant in the study of discrete groups, geometric group theory, and number theory. An approximate group can be thought of as a structure that shares some properties with groups but does not necessarily satisfy all group axioms in a strict sense.

In group theory, the Artin transfer is a specific homomorphism associated with a certain class of groups called "finite groups." More specifically, it is related to the study of group extensions and the relationships between a group and its normal subgroups. The Artin transfer is particularly relevant in the context of modular representation theory and the representation theory of finite groups of Lie type, as well as in the study of central extensions and cohomology.

The Baby-step Giant-step algorithm is a mathematical method used for solving the discrete logarithm problem in a group.

The Banach–Tarski paradox is a theorem in set-theoretic geometry that demonstrates a counterintuitive property of infinite sets. Formulated by mathematicians Stefan Banach and Alfred Tarski in 1924, the paradox states that it is possible to take a solid ball in three-dimensional space, decompose it into a finite number of disjoint non-overlapping pieces, and then reassemble those pieces using only rotations and translations to create two identical copies of the original ball.

Bass–Serre theory is a branch of algebraic topology that studies the relationships between groups and their actions on trees (in a combinatorial sense). Developed by mathematicians Hyman Bass and Jean-Pierre Serre in the 1960s, the theory provides a framework for understanding certain types of groups, particularly finitely generated groups that can be decomposed in terms of simpler pieces.

Bender's method is a term often used in the context of numerical analysis, particularly in relation to solving differential equations and related mathematical problems. Specifically, it refers to a type of numerical scheme used for approximating the solutions of boundary value problems. One notable application of Bender's method is in the numerical solution of ordinary differential equations (ODEs) and partial differential equations (PDEs). The method is typically suited for problems where the solution can exhibit sharp gradients or discontinuities.

The Bianchi groups are a class of groups that arise in the context of hyperbolic geometry and algebraic groups. Specifically, they are related to the modular group of lattices in hyperbolic space. The Bianchi groups can be defined as groups of isometries of hyperbolic 3-space \(\mathbb{H}^3\) that preserve certain algebraic structures. More concretely, the Bianchi groups are associated with imaginary quadratic number fields.

The term "bicommutant" arises in the context of operator algebras and functional analysis, particularly in the study of von Neumann algebras.

The term "Bimonster group" refers to a specific mathematical construct in the field of group theory, particularly in the study of finite groups and modular functions. The Bimonster group is a central extension of the Monster group, which is the largest of the sporadic simple groups. The Bimonster can be described as a group that involves two copies of the Monster group, and it has connections to various areas in mathematics, such as number theory and algebra.

A Bol loop is a type of algebraic structure that generalizes the concept of a group. Specifically, a Bol loop is a non-empty set \( L \) equipped with a binary operation that satisfies certain properties reminiscent of a group but without requiring the existence of an identity element or the inverse for every element.

The Burnside problem is a question in the field of group theory, a branch of abstract algebra. Named after the mathematician William Burnside, the problem essentially asks whether a group with a finite number of orbits under a given group action must necessarily be finite.

A CN-group, or **Cohen–Nonstandard Group**, is a type of mathematical structure in the field of group theory, particularly in the realm of non-standard analysis and model theory. It is a group that can be constructed using certain properties or models of set theory, often involving the use of Cohen forcing or related techniques.

The Caesar cipher is a simple and widely known encryption technique used in cryptography. Named after Julius Caesar, who reportedly used it to communicate with his generals, this cipher is a type of substitution cipher where each letter in the plaintext is 'shifted' a certain number of places down or up the alphabet. For example, with a shift of 3: - A becomes D - B becomes E - C becomes F - ...

In group theory, the **center** of a group \( G \), often denoted as \( Z(G) \), is defined as the set of elements in \( G \) that commute with every other element in the group.

A character group typically refers to a collection or set of characters that share certain characteristics or properties, often used in various contexts including literature, psychology, gaming, and social dynamics. Here are a few interpretations of the term: 1. **Literature and Media**: In storytelling, a character group can refer to a cast of characters that interact within a narrative. This can include protagonists, antagonists, and supporting characters who may have different roles, motivations, and relationships.

A **character table** is a mathematical tool used in the field of group theory, a branch of abstract algebra. It provides a compact way to represent the irreducible representations of a finite group. The character table of a group includes the following key components: 1. **Irreducible Representations**: Each row of the character table corresponds to an irreducible representation (a representation that cannot be decomposed into smaller representations) of the group.

In mathematics, particularly in the field of group theory and algebra, a class automorphism refers to a specific type of automorphism of a group, particularly in the context of a class of structures, such as groups or rings. ### Definition of Automorphism An **automorphism** is an isomorphism from a mathematical structure to itself. In simpler terms, it's a bijective (one-to-one and onto) mapping of a structure that preserves the operations defined on that structure.

In object-oriented programming, a class function (also known as a class method) is a method that is associated with a class rather than with instances of the class. This concept is most commonly found in languages like Python, Java, and C++, where you can define methods that act on the class itself rather than on individual objects. ### Key Characteristics of Class Functions: 1. **Binding to Class**: Class functions are called on the class itself rather than an instance of the class.

Cohomological dimension is a concept from algebraic topology, algebraic geometry, and homological algebra that relates to the size of a space or algebraic object as measured by its cohomology groups. It serves as a measure of the "complexity" of a topological space or algebraic structure in terms of the ability to compute its cohomology.

The commutator subgroup of a group \( G \), often denoted as \( [G, G] \) or sometimes as \( G' \), is a specific subgroup of \( G \) that captures the "non-abelian" structure of the group. It is constructed using the commutators of elements from \( G \).

In group theory, the term "complement" can refer to a few different concepts depending on the context, but it is often associated with subgroup theory.

In the context of algebraic geometry and representation theory, a **complex reflection group** is a specific type of symmetry group that arises in the study of regular polytopes and their symmetries, particularly in complex vector spaces. Formally, a complex reflection group is defined as a finite group generated by complex reflections.

In group theory, the term "component" can refer to various concepts depending on the context. However, one common usage pertains to the component of a group element in a topological or algebraic sense. 1. **Connected Components in Topological Groups**: In the context of topological groups, the component of a group element \( g \) refers to the connected component of the identity element that contains \( g \).

In group theory, a **conjugacy class** is a fundamental concept that helps understand the structure of a group. Given a group \( G \) and an element \( g \in G \), the conjugacy class of \( g \) is the set of elements in \( G \) that can be obtained by conjugating \( g \) with each element of \( G \).

In group theory, a branch of abstract algebra, the concept of a conjugacy class and the associated conjugacy class sum are important for understanding the structure of a group. ### Conjugacy Class A **conjugacy class** of an element \( g \) in a group \( G \) is the set of elements that can be obtained by conjugating \( g \) by all elements of \( G \).

The conjugacy problem is a well-studied question in the field of group theory, a branch of abstract algebra. Specifically, it pertains to determining whether two elements in a group are conjugate to each other.

In the context of isometries in Euclidean space, conjugation refers to the operation that modifies an isometry by another isometry, often to understand how certain properties change under transformations. An isometry is a distance-preserving transformation, which can include translations, rotations, reflections, and glide reflections. In Euclidean space, we can represent isometries using linear transformations (matrices) and translations (vectors).

In group theory, the term "core" can refer to a specific concept related to the notion of a normal subgroup. The core of a subgroup \( H \) of a group \( G \), often denoted as \( \text{Core}_G(H) \), is defined as the largest normal subgroup of \( G \) that is contained within \( H \).

In group theory, which is a branch of abstract algebra, a **coset** is a concept used to describe a way of partitioning a group into smaller, equally structured subsets. Cosets arise when considering a subgroup within a larger group.

The Cremona group is a fundamental concept in algebraic geometry, specifically regarding the study of rational transformations in projective space. In particular, the Cremona group \( \text{Cr}(n) \) refers to the group of birational transformations of \( \mathbb{P}^n \), the \( n \)-dimensional projective space over a field (often taken to be the complex numbers or other algebraically closed fields).

De Sitter invariant special relativity refers to a theoretical framework that extends the principles of special relativity to include de Sitter space, which is a model of spacetime that includes a positive cosmological constant. This framework can be seen as an extension of the usual flat Minkowski spacetime of special relativity, incorporating the geometric properties associated with de Sitter space, which has a constant positive curvature.

A Dedekind group is a specific type of group in the field of abstract algebra, characterized by certain structural properties. The most common definition is that a Dedekind group is a group in which every subgroup is normal. This means that for any subgroup \( H \) of a Dedekind group \( G \), the condition \( gHg^{-1} = H \) holds for every element \( g \) in \( G \).

The Demushkin group, named after the Russian mathematician Dmitry Demushkin, refers to a class of groups that arise in the study of group theory, particularly in the area of pro-p groups. A pro-p group is a class of topological groups that can be understood as inverse limits of finite groups whose orders are powers of a prime \( p \).

In group theory, a descendant tree is a graphical representation used to illustrate the structure of a group, particularly when considering subgroup relationships and the generating processes of those subgroups. It typically represents the idea of iteratively forming subgroups by considering the set of all possible subgroups generated by a given subgroup. ### Key Concepts: 1. **Group**: A set \( G \) equipped with a binary operation that satisfies the group axioms (closure, associativity, identity element, and invertibility).

In the context of group theory, particularly in the study of algebraic groups and Lie groups, a diagonal subgroup is typically a subgroup that is constructed from the diagonal elements of a product of groups. For example, consider the direct product of two groups \( G_1 \) and \( G_2 \).

In group theory, the direct sum of groups is a construction that allows one to combine two or more groups into a new group in a way that preserves their individual structures. The direct sum is often denoted by the symbol \(\oplus\) or sometimes as a product, depending on the context.

In group theory, a double coset is a concept associated with a group acting on itself in a specific way. More formally, if \( G \) is a group and \( H \) and \( K \) are two subgroups of \( G \), the double coset of \( H \) and \( K \) with respect to an element \( g \in G \) is denoted by \( HgK \).

The term "Double group" can refer to different concepts in different contexts, but most commonly, it is associated with the fields of group theory in mathematics, particularly in the context of symmetry and crystallography. Here are a couple of interpretations: 1. **Double Group in Group Theory**: In the context of group theory, a double group is a mathematical construct that arises to account for certain symmetries.

The Ehlers group is a mathematical concept used primarily in the field of differential geometry and general relativity. Specifically, it refers to a group of transformations that preserve certain structures in the context of spacetime symmetries. In general relativity, the Ehlers group is associated with the symmetry properties of solutions to Einstein's field equations, particularly in the study of stationary spacetimes.

An elliptic curve is a type of mathematical structure that has important applications in various fields, including number theory, cryptography, and algebraic geometry. Formally, an elliptic curve is defined as the set of points \( (x, y) \) that satisfy a specific type of equation in two variables.

In the context of machine learning and natural language processing, the term "embedding problem" can refer to several related concepts, primarily revolving around the challenge of representing complex data in a form that can be effectively processed by algorithms. Here are some key aspects: 1. **Embedding Vectors**: In machine learning, "embedding" typically refers to the transformation of high-dimensional data into a lower-dimensional vector space. This is crucial for enabling efficient computation and understanding relationships between data points.

The Engel Group typically refers to a series of companies or divisions under the Engel brand, which is known for manufacturing injection molding machines and automation technology, primarily for the plastic processing industry. Engel is an international company based in Austria that provides solutions for various applications, including automotive, packaging, medical technology, and consumer goods.

The Engel identity is an important concept in the context of consumer theory in economics, particularly related to how income affects consumption patterns. It is named after the German statistician Ernst Engel. The Engel identity states that for a given good or a set of goods, the share of total income spent on that good (or those goods) is a function of income.

The term "groups" can refer to various contexts, including social organizations, mathematical structures, and classification of entities. Here are examples from different domains: ### Social Groups 1. **Friendship Groups**: A circle of friends who meet regularly. 2. **Family Groups**: Extended families that gather for events or holidays. 3. **Work Teams**: Employees collaborating on projects in a workplace.

The term "Fibonacci group" can refer to different contexts depending on the field of study.

A finitely generated group is a group \( G \) that can be generated by a finite set of elements. More formally, there exists a finite set of elements \( \{ g_1, g_2, \ldots, g_n \} \) in \( G \) such that every element \( g \in G \) can be expressed as a finite combination of these generators and their inverses.

Finiteness properties of groups refer to various conditions that describe the size and structure of groups in terms of the existence or non-existence of certain substructures. These properties often deal with group actions, representations, and how a group can be constructed or decomposed in terms of its subgroups.

In the context of Euclidean space, an isometry is a transformation that preserves distances. This means that if you have two points \( A \) and \( B \) in Euclidean space, an isometric transformation \( T \) will maintain the distance between these points, i.e., \( d(T(A), T(B)) = d(A, B) \), where \( d \) denotes the distance function.

In group theory, "formation" refers to a class of groups that share certain properties, particularly related to their behavior with respect to subgroup structure, normal subgroups, and composition factors. Formations are typically defined in the context of specific conditions that a group must satisfy to belong to the formation. The most common way to define a formation is through the concept of a **variety** of groups (a class of groups defined by a set of group identities) that is closed under certain operations.

The Frattini subgroup is an important concept in group theory, particularly in the study of finite groups. It is defined as the subgroup of a group \( G \) that is generated by all the non-generators of \( G \). Specifically, it has a few equivalent characterizations: 1. **Definition**: The Frattini subgroup \( \Phi(G) \) of a group \( G \) is the intersection of all maximal subgroups of \( G \).

In the context of algebra, particularly in representation theory and module theory, a **G-module** is a module that is equipped with an action by a group \( G \). Specifically, if \( G \) is a group and \( M \) is a module over a ring \( R \), a \( G \)-module is a set \( M \) together with a group action of \( G \) on \( M \) that is compatible with the operation of \( M \).

The Generalized Dihedral Group, often denoted \( \text{GD}(n) \) or \( D_n^* \), is a group that generalizes the properties of the traditional dihedral group. The dihedral group \( D_n \) is the group of symmetries of a regular polygon with \( n \) sides, and it includes both rotations and reflections. It has the order \( 2n \) (i.e.

A generating set of a group is a subset of the group's elements such that every element of the group can be expressed as a combination of the elements in the generating set using the group's operation (e.g., multiplication, addition).

Geometric group theory is a branch of mathematics that studies the connections between group theory and geometry, particularly through the lens of topology and geometric structures. It emerged in the late 20th century and has since developed into a rich area of research, incorporating ideas from various fields including algebra, topology, and geometry. Key concepts in geometric group theory include: 1. **Cayley Graphs**: These are graphical representations of groups that illustrate the group's structure.

The \((2,3,7)\) triangle group, denoted as \(\Delta(2,3,7)\), is a type of discrete group that arises in the study of hyperbolic geometry and can be constructed as a group of isometries of hyperbolic space.

The Adian–Rabin theorem is a result in the field of mathematical logic, specifically in the area of decidability and the theory of algebraic structures. It addresses the properties of certain classes of roots of equations and relies on concepts from algebra and logic. In basic terms, the theorem states that for any given sequence of rational numbers, it is possible to find a computably enumerable sequence of algebraic numbers that has roots within those rational numbers.

Asymptotic dimension is a concept from geometric topology and metric geometry that provides a way to measure the "size" or "dimension" of a metric space in a manner that is sensitive to the space's large-scale structure. It was introduced by the mathematicians J. M. G. B. Connes and more extensively developed by others in the context of spaces that arise in analysis, algebra, and topology.

The Curve Complex is a mathematical structure used in the field of low-dimensional topology, particularly in the study of surfaces. It provides a combinatorial way to study the mapping class group of a surface, which is the group of isotopy classes of homeomorphisms of the surface.

In mathematics, particularly in the field of group theory, a **discrete group** is a type of group that is equipped with the discrete topology. To understand this concept, let's break it down: 1. **Group**: A group is a set \( G \) along with an operation \( \cdot \) (often just denoted by juxtaposition) that satisfies four fundamental properties: closure, associativity, identity, and the existence of inverses.

Flexagon is a term that can refer to a few different concepts, depending on the context. However, it is most commonly recognized in the following ways: 1. **In Mathematics**: A flexagon is a type of flexible polygonal structure that can be manipulated to reveal different faces.

The term "free factor complex" often arises in the context of group theory, particularly in the study of free groups and their actions. A free group is a group that has a basis such that every element can be uniquely expressed as the product of finitely many basis elements and their inverses.

In the context of group theory, specifically in the study of automorphisms of algebraic structures, a **fully irreducible automorphism** generally refers to a certain type of automorphism of a free group or a free object in category theory.

A Følner sequence is a concept from the field of mathematical analysis, particularly in ergodic theory and group theory. It is named after the mathematician Ernst Følner. A Følner sequence provides a way to study the asymptotic behavior of actions of groups on sets and is often used in the context of amenable groups.

A geometric group action is a specific type of action by a group on a geometric space, which can often be thought of in terms of symmetries or transformations of that space. More formally, if we have a group \( G \) and a geometric object (often a topological space or manifold) \( X \), a geometric group action is defined when \( G \) acts on \( X \) in a way that respects the structure of \( X \).

A "graph of groups" is a combinatorial and algebraic structure that can be used to study groups, particularly in the context of group theory and geometric topology. It is a way to construct larger groups from smaller ones by specifying how they are connected through a graph. ### Components of a Graph of Groups: 1. **Graph**: A graph \( G \) consists of vertices (also called nodes) and edges connecting them.

The Gromov boundary is a concept in geometric topology, particularly in the study of metric spaces, especially those that are geodesic and hyperbolic. It is used to analyze the asymptotic behavior of spaces and to understand their large-scale geometry. More formally, the Gromov boundary can be defined for a proper geodesic metric space. A metric space is considered proper if every closed ball in the space is compact.

The Grushko theorem is a result in the field of group theory, particularly concerning free groups and their subgroups. It provides a criterion to establish whether a given group is free and helps characterize the structure of free groups.

The Haagerup property, also known as being "exact," refers to a specific geometric property of certain groups or von Neumann algebras in the context of functional analysis and noncommutative geometry. It is named after Danish mathematician Uffe Haagerup, who first introduced the concept in the context of von Neumann algebras.

The Kurosh subgroup theorem is a result in group theory, specifically concerning the structure of subgroups of a given group. It provides a description of the subgroups of a free group or a subgroup of a free group.

The mapping class group of a surface is a fundamental concept in the field of algebraic topology and differential geometry. Given a surface \( S \), the mapping class group, denoted \( \mathrm{Mod}(S) \), consists of equivalence classes of orientation-preserving homeomorphisms of the surface modulo the action of homotopy.

In the context of mathematics and particularly in set theory or function theory, "Out(Fn)" is not a widely recognized standard notation or term. However, it may relate to various concepts depending on what "Fn" specifically denotes. If "Fn" represents a function, for instance, "Out(Fn)" could refer to the output of that function.

In mathematics, "outer space" typically refers to a certain type of geometric space associated with free groups and their actions. The most common reference is to "Outer space" denoted as \( \mathcal{O}(F_n) \), which is the space of marked metric graphs that correspond to the free group \( F_n \) of rank \( n \).

Quasi-isometry is a concept in metric geometry and geometric group theory that provides a way to compare metric spaces.

A relatively hyperbolic group is a type of group in geometric group theory that generalizes the concept of hyperbolic groups. A group \( G \) is said to be relatively hyperbolic with respect to a collection of subgroups \( \mathcal{P} \) if the asymptotic geometry of \( G \) behaves somewhat like that of a hyperbolic group, but it can include additional structure provided by the subgroups in \( \mathcal{P} \).

The term "Rips machine" could refer to several things, but in a common context, it often relates to a "Rips" machine used for a specific purpose in various industries. Here are some possibilities: 1. **Rips Software**: In computational topology, Rips complexes are used to study metric spaces. A machine or software that implements Rips complexes allows researchers to analyze the structure and properties of data using topological methods.

Stallings' theorem concerns the structure of finitely generated groups in relation to their ends. In topology, the "ends" of a space can intuitively be understood as the number of "directions" in which the space can be infinitely extended. For groups, ends are related to how a group's Cayley graph behaves at infinity.

Subgroup distortion refers to a phenomenon in which the characteristics, behaviors, or identities of individuals within a subgroup of a larger population are misrepresented or misunderstood, often due to stereotypes or biases. This can occur in various contexts, including social groups, organizational settings, and research.

The Thurston boundary is a concept from the field of topology, particularly in the study of 3-manifolds. More specifically, it refers to a boundary that arises in the context of 3-dimensional hyperbolic geometry and is used in the classification of 3-manifolds. In general terms, the Thurston boundary often arises in relation to the concept of a compactification of a space.

As of my last knowledge update in October 2023, "Ultralimit" could refer to various concepts depending on the context in which it is used. However, there wasn't a widely recognized or specific definition for "Ultralimit" in major fields such as technology, science, or popular culture.

The Weyl distance function is a mathematical tool used in the field of differential geometry and the study of Riemannian manifolds. It is particularly important when analyzing the geometry of spaces that have different curvature properties. The concept is closely associated with Weyl's notion of conformal equivalence. In a more formal sense, the Weyl distance function can be defined within the context of Riemannian geometry.

The Švarc–Milnor lemma is a result in differential geometry and algebraic topology, particularly concerning the relationship between the topology of a space and the geometry of its covering spaces. It is named after mathematicians David Švarc and John Milnor.

A glossary of group theory includes key terms, definitions, and concepts that are fundamental to understanding group theory, a branch of abstract algebra. Here are some essential terms and their meanings: 1. **Group**: A set \( G \) equipped with a binary operation \( \cdot \) that satisfies four properties: closure, associativity, identity element, and invertibility.

The Grigorchuk group is an important example of a group in geometric group theory and is particularly known for its striking properties. It was introduced by the Mathematician Rostislav Grigorchuk in 1980 and is often classified as a "locally finitely presented" group.

Group extension is a concept in group theory, a branch of abstract algebra. It refers to the process of creating a new group from a known group by adding new elements that satisfy certain properties related to the original group. More formally, it describes a way to construct a group \( G \) that contains a normal subgroup \( N \) and a quotient group \( G/N \).

Groups of Lie type are a class of algebraic groups that can be associated with simple Lie algebras and are defined over finite fields. They play a significant role in the theory of finite groups, particularly in the classification of finite simple groups. The concept of groups of Lie type arises from the representation theory of Lie algebras over fields, especially over finite fields.

Group representation is a concept from the field of abstract algebra and representation theory, which studies how groups can be represented by matrices and how their elements can act on vector spaces. Essentially, a group representation provides a way to express abstract group elements as linear transformations (or matrices) acting on a vector space. ### Key Concepts: 1. **Group**: A set equipped with an operation that satisfies four properties: closure, associativity, the existence of an identity element, and the existence of inverses.

Group structure and the Axiom of Choice are concepts from different areas of mathematics: group theory and set theory, respectively. Here’s a brief overview of both concepts: ### Group Structure A group is a fundamental algebraic structure in mathematics, particularly in the field of group theory.

Hall's identity is a mathematical result related to the theory of partitions and combinatorial identities. Specifically, it provides a relationship involving binomial coefficients, which can be viewed through the lens of combinatorial enumeration. The identity states that for any non-negative integer \( n \): \[ \sum_{k=0}^{n} (-1)^k \binom{n}{k} (n-k)^m = (-1)^n \binom{m}{n} n!

The Hanna Neumann Conjecture is a hypothesis in the field of group theory, specifically concerning the relationship between the rank of a group and the ranks of its subgroups.

The Higman group, often denoted as \( \text{H} \), is a notable example of a group in the field of group theory, particularly in the area of infinite groups. It was constructed by Graham Higman in the 1950s as an example of a finitely generated group that is not finitely presented. The Higman group can be defined using a particular way of organizing its generators and relations.

The Higman–Sims asymptotic formula is a result in the area of group theory and combinatorics, particularly relating to the structure of finitely generated groups and specifically the growth rates of certain groups. Named after Graham Higman and Charles Sims, this formula provides an asymptotic estimate for the number of groups of a given order.

Group theory is a branch of mathematics that studies algebraic structures known as groups, which encapsulate the concept of symmetry. The history of group theory traces its development through several key milestones and figures. ### Early Foundations (17th - 18th Century) - **Symmetry and Permutations**: The notion of symmetry in geometry and transformations can be traced back to the work of mathematicians like René Descartes and Isaac Newton.

In mathematics, particularly in the field of complex analysis, the term "holomorph" typically refers to a function that is holomorphic. A holomorphic function is a complex function that is defined on an open subset of the complex plane and is differentiable at every point in its domain with respect to the complex variable.

The concept of a **Homeomorphism group** arises in the field of topology, which is the study of the properties of space that are preserved under continuous transformations. Let's break down what a homeomorphism is and then define the homeomorphism group. ### Homeomorphism A **homeomorphism** is a special type of function between two topological spaces.

In the context of mathematics, specifically in the field of group theory and algebraic structures, an "IA automorphism" generally refers to an automorphism of a group that fixes a certain subgroup, or more specifically, it preserves certain structural properties of the group. The terms "IA" typically stands for "Inner Automorphism," which refers to automorphisms that can be expressed as conjugation by an element of the group.

An idempotent measure refers to a type of measure in the context of mathematical analysis, particularly in the fields of functional analysis and probability theory, where the concept of idempotence plays a key role. In general terms, something is considered idempotent if an operation can be applied multiple times without changing the result beyond the initial application.

The Index Calculus algorithm is a classical algorithm used for solving the discrete logarithm problem in certain algebraic structures, such as finite fields and elliptic curves. The discrete logarithm problem can be described as follows: given a prime \( p \), a generator \( g \) of a group \( G \), and an element \( h \in G \), the goal is to find an integer \( x \) such that \( g^x \equiv h \mod p \).

In group theory, the index of a subgroup is a concept that helps to measure the "size" of the subgroup in relation to the larger group it belongs to. Specifically, if \( G \) is a group and \( H \) is a subgroup of \( G \), the index of \( H \) in \( G \), denoted as \( [G : H] \), is defined as the number of distinct left cosets of \( H \) in \( G \).

Induced characters refer to representations of a group that arise from the representation of a subgroup. In the context of representation theory—an area of mathematics that studies abstract algebraic structures through linear transformations—induced characters are a way to construct new representations of a group via a subgroup.

Induced representation is a concept from representation theory in mathematics, particularly in the study of group theory. It allows one to construct a representation of a larger group from a representation of a subgroup. To understand induced representations, consider the following key ideas: 1. **Groups and Representations**: A group is a mathematical structure consisting of a set of elements equipped with an operation that satisfies certain axioms (closure, associativity, identity, and invertibility).

An **inner automorphism** is a specific type of automorphism of a group that arises from the structure of the group itself. In group theory, an automorphism is a bijective homomorphism from a group to itself, meaning it is a structure-preserving map that reflects the group's operations. An inner automorphism can be defined as follows: Let \( G \) be a group and let \( g \) be an element of \( G \).

Invariant decomposition is a mathematical technique used primarily in the field of dynamical systems, control theory, and related areas. The essence of invariant decomposition is to break down a complex system into simpler, more manageable components or subsystems that can be analyzed independently. These components remain invariant under certain transformations or conditions, which often simplifies both their analysis and control.

In group theory, a **lattice of subgroups** refers to the structure that can be formed by the collection of subgroups of a given group, ordered by the inclusion relation. Specifically, it involves the following key concepts: 1. **Subgroups**: A subgroup is a subset of a group that is also a group under the same operation.

The Leinster Group is a geological formation located in eastern Ireland, primarily in the province of Leinster. It consists of a sequence of rocks that were formed in the late Paleozoic era, specifically during the Carboniferous period. The group is notable for its varied sedimentary deposits, which include sandstones, mudstones, and limestones.

The "length" function is commonly found in various programming languages and environments, and it is used to determine the number of elements in a data structure, such as a string, array, list, or other collections. Here's a brief overview of how the length function is used in some popular programming languages: 1. **Python**: - In Python, the `len()` function is used to return the number of items in an object.

The character tables for chemically important 3D point groups provide crucial information about the symmetry properties of molecules and their corresponding vibrational modes. Below is a list of the most common point groups along with their character tables: ### Character Tables for 3D Point Groups 1.

The list of finite simple groups is a comprehensive classification of finite groups that cannot be decomposed into simpler groups. A finite simple group is defined as a nontrivial group whose only normal subgroups are the trivial subgroup and the group itself. Finite simple groups can be categorized into several families: 1. **Cyclic Groups of Prime Order**: These are groups of the form \( \mathbb{Z}/p\mathbb{Z} \) for a prime \( p \).

Group theory is a branch of mathematics that studies the algebraic structures known as groups. Below is a list of topics commonly covered in group theory: 1. **Basic Definitions** - Group (definition, binary operation) - Subgroup - Cosets (left and right) - Factor groups (quotient groups) - Order of a group - Order of an element 2.

The Lorentz group is a fundamental group in theoretical physics that describes the symmetries of spacetime in special relativity. Named after the Dutch physicist Hendrik Lorentz, it consists of all linear transformations that preserve the spacetime interval between events in Minkowski space. In mathematical terms, the Lorentz group can be defined as the set of all Lorentz transformations, which are transformations that can be expressed as linear transformations of the coordinates in spacetime that preserve the Minkowski metric.

The Lyndon–Hochschild–Serre spectral sequence is a tool in algebraic topology and homological algebra that arises in the context of group cohomology and the study of group extensions. It provides a method for computing the cohomology of a group \( G \) by relating it to the cohomology of a normal subgroup \( N \) and the quotient group \( G/N \).

A magnetic space group is a mathematical description that combines the symmetry properties of crystal structures with the additional symmetrical aspects introduced by magnetic ordering. In crystallography, a space group describes the symmetrical arrangement of points in three-dimensional space. When we consider magnetic materials, the arrangement of magnetic moments (spins) within the crystal lattice can also possess symmetry that must be accounted for.

Maria Wonenburger is a notable Spanish mathematician known for her work in the field of mathematics, particularly in the areas of algebra and geometry. She made significant contributions to the study of algebraic structures, particularly in relation to group theory and algebraic topology. Wonenburger's work has been influential in advancing mathematical knowledge and understanding in these areas. In addition to her research contributions, she has also been recognized for her efforts in promoting mathematics, especially encouraging women to pursue careers in the field.

"Measurable acting group" does not appear to refer to a widely recognized term or concept in the fields of acting, performance, or any related discipline as of my last update in October 2023. It’s possible that it could refer to a specific group or project, perhaps one that incorporates methods of measuring performance or impact in acting.

A measurable group is a concept from the field of measure theory, a branch of mathematics that deals with the formalization of notions such as size, area, and volume in more complex settings. Specifically, a measurable group is a group equipped with a measure that allows for the integration and differentiation of functions defined on that group.

The Mennicke symbol is an important concept in the study of algebraic K-theory, particularly in the area of the K-theory of fields. It is named after the mathematician H. Mennicke, and it arises in the context of understanding the links between different classes of algebraic structures, particularly in the context of quadratic forms and their associated bilinear forms. In more technical terms, the Mennicke symbol is used to represent certain equivalence classes of quadratic forms over a field.

The modular group is a fundamental concept in mathematics, particularly in the fields of algebra, number theory, and complex analysis. It is defined as the group of 2x2 integer matrices with determinant equal to 1, modulo the action of integer linear transformations on the complex upper half-plane.

A **Moufang loop** is a structure in the field of algebra, specifically in the study of non-associative algebraic systems. A Moufang loop is defined as a set \( L \) equipped with a binary operation (often denoted by juxtaposition) that satisfies the following Moufang identities: 1. \( x(yz) = (xy)z \) 2. \( (xy)z = x(yz) \) 3.

A Moufang set is a concept from the field of mathematics, specifically in the context of algebra and geometry. It is related to the study of certain types of algebraic structures that exhibit properties reminiscent of groups but without necessarily adhering to all the group axioms.

A **multiplicative character** is a type of mathematical function used in number theory, particularly in the context of Dirichlet characters and L-functions. Specifically, a multiplicative character is a homomorphism from the group of non-zero integers under multiplication to a finite abelian group, such as the group of complex numbers of modulus one.

In group theory, a branch of abstract algebra, a **no small subgroup** refers to a specific property of groups that have no nontrivial subgroups of a small size compared to the group itself. More formally, a group \( G \) is said to be a "no small subgroup" group if it does not have any nontrivial subgroups whose order is less than a certain threshold relative to the order of \( G \).

In mathematics, a "norm" in the context of a group typically refers to a concept from group theory, specifically related to the structure of groups and their subgroups. However, the term "norm" can have different meanings depending on the context. 1. **Subgroup Norm**: In the context of finite groups, the term "norm" can refer to the **normalizer** of a subgroup.

In group theory, the concept of normal closure is related to the idea of normal subgroups. Given a group \( G \) and a subset \( H \) of \( G \), the normal closure of \( H \) in \( G \), denoted by \( \langle H \rangle^G \) or sometimes \( \langle H \rangle^n \), is the smallest normal subgroup of \( G \) that contains the set \( H \).