Abelian sections of the symmetric groups with respect to their index

Luca Sabatini

Published 2021-07-13Version 1

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 $|H/H'| \leq \exp \left( \frac{\alpha \cdot \log|G:H|}{\log \log |G:H|} \right)$. 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.