The expected number of elements to generate a finite group with $d$-generated Sylow subgroups

TL;DR

This paper derives bounds on the expected number of random elements needed to generate finite groups with Sylow subgroups generated by d elements, providing explicit formulas involving the Riemann zeta function.

Contribution

It introduces explicit bounds on the expected number of elements to generate certain finite groups, connecting group generation with special functions like the Riemann zeta function.

Findings

For groups with Sylow subgroups generated by d elements, e(G) ≤ d + κ.

Explicit value of κ involving the Riemann zeta function, proven to be optimal.

Specific bounds for permutation groups, with a special case for S_3.

Abstract

Given a finite group $G,$ let $e(G)$ be expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found. If all the Sylow subgroups of $G$ can be generated by $d$ elements, then $e(G)\leq d+\kappa$ with $\kappa \sim 2.75239495.$ The number $\kappa$ is explicitly described in terms of the Riemann zeta function and is best possible. If $G$ is a permutation group of degree $n,$ then either $G=S_3$ and $e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor+\kappa^*$ with $\kappa^* \sim 1.606695.$