On finite groups whose Sylow subgroups have a bounded number of generators

TL;DR

This paper investigates finite groups with Sylow subgroups generated by a limited number of elements, establishing bounds on their structure and related subgroup indices, with explicit results for soluble groups.

Contribution

It proves that such groups have a non-nilpotent image with a characteristic kernel of bounded index, advancing understanding of their subgroup structure.

Findings

Existence of a non-nilpotent image with bounded index

Bounded index of the Frattini subgroup in terms of d and p

Explicit bounds for soluble groups

Abstract

Let G be a finite non-nilpotent group such that every Sylow subgroup of G is generated by at most d elements, and such that p is the largest prime dividing |G|. We show that G has a non-nilpotent image G/N, such that N is characteristic and of index bounded by a function of d and p. This result will be used to prove that the index of the Frattini subgroup of G is bounded in terms of d and p. Upper bounds will be given explicitly for soluble groups.