Subgroups of symmetric groups: enumeration and asymptotic properties

TL;DR

The paper resolves longstanding questions about the enumeration of subgroups of the symmetric group ${\mathrm{S}}_n$ and the structure of random subgroups. It develops a comprehensive toolkit based on Goursat's lemma, subdirect products of finite $p$-groups, and refined bounds on homomorphisms and normal generators, then applies a bounded-orbit reduction to enable sharp asymptotics. The main contributions include the proof of Pyber's conjecture with ${\mathrm{Sub}}({\mathrm{S}}_n) = 2^{n^2/16+o(n^2)}$ and tight bounds for ${\mathrm{Sub}}_p({\mathrm{S}}_n)$, as well as results describing the distribution of nilpotent and abelian sections among random subgroups. The results have implications for computational group theory, Galois theory, and combinatorial enumeration (e.g., transitive graphs), and they demonstrate that Kantor's conjecture about random subgroups of ${\mathrm{S}}_n$ being nilpotent fails in the infinite family of cases constructed. Overall, the work provides a detailed asymptotic map of subgroup growth in symmetric groups and reveals nuanced probabilistic behavior of random permutation groups.

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 $1$.

Theorems & Definitions (107)

• Theorem 1
• Theorem 2
• Theorem 3
• Conjecture : Kantor's conjecture
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Lemma 2.1
• Definition 2.2