Abelian sections of the symmetric groups with respect to their index

Abstract

We show the existence of an absolute constant $\alpha >0$ such that, for every $k\ge 3$, $G:=\mathrm{Sym}(k)$, and for every $H⩽G$ of index at least $3$, one has $|H/[H,H]|\le |G:H{|}^{\alpha /\log \log |G:H|}$. 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.

6 pages

Cited by in corpus (1)

• The growth of abelian sections