The length of mixed identities for finite groups

TL;DR

This paper establishes bounds on the length of mixed identities in finite groups, showing that groups with no short non-trivial mixed identities are almost simple with Lie type socles, and provides specific results for classical groups.

Contribution

It introduces the concept of length of mixed identities, proves existence of a universal constant for almost simple groups, and derives rank-independent bounds for classical groups based on field size.

Findings

Existence of a constant c such that groups with no short mixed identities are almost simple.

Rank-independent bounds for the length of mixed identities in classical groups.

Results specific to groups with socles PSL, PSp, PΩ, and PSU.

Abstract

We prove that there exists a constant $c>0$ such that any finite group having no non-trivial mixed identity of length $\leq c$ is an almost simple group with a simple group of Lie type as its socle. Starting the study of mixed identities for almost simple groups, we obtain results for groups with socle ${\rm PSL}_n(q)$, ${\rm PSp}_{2m}(q)$, ${\rm P \Omega}_{2m-1}^\circ(q)$, and ${\rm PSU}_n(q)$ for a prime power $q$. For such groups, we will prove rank-independent bounds for the length of a shortest non-trivial mixed identity, depending only on the field size $q$.