Almost all alternating groups are invariably generated by two elements of prime order

TL;DR

This paper proves that nearly all alternating groups up to a certain size are generated by two prime order elements, answering a question about their invariable generation.

Contribution

It provides a quantitative proof that almost all alternating groups are invariably generated by two elements of prime order, extending understanding of their generation properties.

Findings

Almost all $A_n$ are generated by two prime order elements

The number of exceptions grows slower than $X ext{exp}(-c( ext{log}X)^{1/2}( ext{log} ext{log}X)^{1/2})$

Answers a question of Guralnick, Shareshian, and Woodroofe

Abstract

We show that for all $n\leq X$ apart from $O(X\exp(-c(\log X)^{1/2}(\log \log X)^{1/2}))$ exceptions, the alternating group $A_n$ is invariably generated by two elements of prime order. This answers (in a quantitative form) a question of Guralnick, Shareshian and Woodroofe.