Invariable generation and the chebotarev invariant of a finite group

TL;DR

This paper establishes tight bounds on the size of invariable generating sets for finite groups, proves that all finite simple groups are invariably generated by two elements, and explores probabilistic aspects of invariable generation.

Contribution

It provides the first tight upper bounds on invariable generating set sizes and answers a key question about random invariable generation in finite groups.

Findings

Tight upper bounds on minimal invariable generating set size

Finite simple groups are invariably generated by two elements

Probabilistic bounds on random invariable generation

Abstract

A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si > for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response to a question in [KZ] we also bound the size of a randomly chosen set of elements of G that is likely to generate G invariably. Along the way we prove that every finite simple group is invariably generated by two elements.