Subgroups of symmetric groups: enumeration and asymptotic properties

Abstract

In this paper, we prove that the symmetric group ${\mathrm{S}}_n$ has $2^{n^2/16+o(n^2)}$ subgroups, settling a conjecture of Pyber from 1993. We also derive asymptotically sharp upper and lower bounds on the number of subgroups of ${\mathrm{S}}_n$ of various kinds, including the number of $p$-subgroups. In addition, we prove a range of theorems about random subgroups of ${\mathrm{S}}_n$. In particular, we prove the surprising result that for infinitely many $n$, the probability that a random subgroup of ${\mathrm{S}}_n$ is nilpotent is bounded away from $1$.