Abstract
AbstractWe show the existence of an absolute constant $$\alpha >0$$
α
>
0
such that, for every $$k \ge 3$$
k
≥
3
, $$G:= \mathop {\mathrm {Sym}}(k)$$
G
:
=
Sym
(
k
)
, and for every $$H \leqslant G$$
H
⩽
G
of index at least 3, one has $$|H/H'| \le |G:H|^{\alpha / \log \log |G:H|}$$
|
H
/
H
′
|
≤
|
G
:
H
|
α
/
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.
Funder
istituto nazionale di alta matematica “francesco severi”
Publisher
Springer Science and Business Media LLC
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