Fibers of automorphic word maps and an application to composition factors

TL;DR

This paper investigates the fibers of automorphic word maps on finite groups, especially simple groups, revealing restrictions on their composition factors based on fiber sizes, with implications for group structure analysis.

Contribution

It introduces a study of automorphic word map fibers and establishes new restrictions on the composition factors of finite groups with large fiber sizes.

Findings

Large fibers exclude certain simple groups as composition factors.

No high-rank classical simple groups can occur as factors under fiber size conditions.

Provides structural restrictions on finite groups based on automorphic word map fibers.

Abstract

In this paper, we study the fibers of "automorphic word maps", a certain generalization of word maps, on finite groups and on nonabelian finite simple groups in particular. As an application, we derive a structural restriction on finite groups $G$ where, for some fixed nonempty reduced word $w$ in $d$ variables and some fixed $\rho\in\left(0,1\right]$, the word map $w_G$ on $G$ has a fiber of size at least $\rho|G|^d$: No sufficiently large alternating group and no (classical) simple group of Lie type of sufficiently high rank can occur as a composition factor of such a group $G$.