Subgroup growth

In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group.^[1]

Let ${\displaystyle G}$ be a finitely generated group. Then, for each integer ${\displaystyle n}$ define ${\displaystyle a_{n}(G)}$ to be the number of subgroups ${\displaystyle H}$ of index ${\displaystyle n}$ in ${\displaystyle G}$. Similarly, if ${\displaystyle G}$ is a topological group, ${\displaystyle s_{n}(G)}$ denotes the number of open subgroups ${\displaystyle U}$ of index ${\displaystyle n}$ in ${\displaystyle G}$. One similarly defines ${\displaystyle m_{n}(G)}$ and ${\displaystyle s_{n}^{\triangleleft }(G)}$ to denote the number of maximal and normal subgroups of index ${\displaystyle n}$, respectively.

Subgroup growth studies these functions, their interplay, and the characterization of group theoretical properties in terms of these functions.

The theory was motivated by the desire to enumerate finite groups of given order, and the analogy with Mikhail Gromov's notion of word growth.