Characteristic Covering Numbers of Finite Simple Groups

TL;DR

This paper proves that for any six non-identity words, their combined images cover all finite simple groups, and introduces general results on characteristic collections and covering numbers with broader applications.

Contribution

It establishes new bounds on covering numbers of finite simple groups using characteristic collections and explores their implications.

Findings

Six non-identity words cover all finite simple groups when combined.

Either a word's sixth power covers all finite simple groups or it is trivial on some groups.

General results on characteristic collections provide new bounds and applications.

Abstract

We show that, if $w_1, \ldots , w_6$ are words which are not an identity of any (non-abelian) finite simple group, then $w_1(G)w_2(G) \cdots w_6(G) = G$ for all (non-abelian) finite simple groups $G$. In particular, for every word $w$, either $w(G)^6 = G$ for all finite simple groups, or $w(G)=1$ for some finite simple groups. These theorems follow from more general results we obtain on characteristic collections of finite groups and their covering numbers, which are of independent interest and have additional applications.