Abelian sections of the symmetric groups with respect to their index

TL;DR

This paper establishes a bound on the size of the abelianization of subgroups of symmetric groups relative to their index, revealing a precise asymptotic behavior and conjecturing its universality.

Contribution

It proves a sharp inequality relating subgroup abelianization size to index in symmetric groups and conjectures its optimality for all large finite groups.

Findings

The inequality is tight for symmetric groups.

The bound depends on the subgroup index and logarithmic factors.

Conjecture: the inequality is optimal for all large finite groups.

Abstract

We show the existence of an absolute constant $\alpha>0$ such that, for every $k \geq 3$, $G:=\mathop{\mathrm{Sym}}(k)$, and for every $H \leqslant G$ of index at least $3$, one has $|H/[H,H]| \leq |G:H|^{\alpha/ \log \log |G:H|}$. This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups.