On non-uniformly simple groups

TL;DR

This paper investigates the property of simple groups where elements can be expressed as products of conjugates, showing that the infinite alternating group is not uniformly simple and analyzing the unboundedness of such expressions.

Contribution

It demonstrates that the infinite alternating group is non-uniformly simple and provides an evaluation of how the expression length grows without bound.

Findings

Infinite alternating group is non-uniformly simple.

The length of expressing elements as conjugate products is unbounded.

Provides quantitative analysis of the unboundedness.

Abstract

Suppose $G$ is a simple group. For any nontrivial elements $g$ and $h$, $g$ can be written as a finite product of conjugates of $h$ or the inverse of $h$. G is called uniformly simple if the length of such an expression is uniformly bounded. We show that the infinite alternating group is non-uniformly simple and evaluate how the length of such an expression is unbounded.