Index of a subgroup

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 \( G \).