A bound on the expected number of random elements to generate a finite group all of whose Sylow subgroups are d-generated

TL;DR

This paper establishes an upper bound on the expected number of random elements needed to generate a finite group with Sylow subgroups that are all d-generated, providing a probabilistic measure of group generation.

Contribution

It introduces a bound on the expected number of random elements required to generate such groups, extending understanding of probabilistic generation in finite groups.

Findings

Expected number of elements needed is at most d + 2.875065

Provides a probabilistic bound applicable to groups with d-generated Sylow subgroups

Enhances theoretical understanding of random generation in finite group theory

Abstract

Assume that all the Sylow subgroups of a finite group $G$ can be generated by $d$ elements. Then the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found, is at most $d+\eta$ with $\eta \sim 2.875065.$