In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group. Let be a finitely generated group. Then, for each integer define to be the number of subgroups of index in . Similarly, if is a topological group, denotes the number of open subgroups of index in . One similarly defines and to denote the number of maximal and normal subgroups of index , respectively. Subgroup growth studies these functions, their interplay, and the characterization of group theoretical properties in terms of these functions.
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