On the finite simple images of free products of finite groups

TL;DR

The paper proves that large finite simple groups can be obtained as images of free products of finite groups, confirming a conjecture by Tamburini and Wilson.

Contribution

It establishes that all sufficiently large finite simple groups are images of free products of two finite groups, and shows they are generated by subgroups isomorphic to these groups.

Findings

Every large finite simple group is an image of a free product of finite groups.

Such groups are generated by subgroups isomorphic to the original finite groups.

The conjecture of Tamburini and Wilson is confirmed.

Abstract

Given nontrivial finite groups $A$ and $B$, not both of order 2, we prove that every finite simple group of sufficiently large rank is an image of the free product $A \ast B$. To show this, we prove that every finite simple group of sufficiently large rank is generated by a pair of subgroups isomorphic to $A$ and $B$. This proves a conjecture of Tamburini and Wilson.