The length of mixed identities for finite groups

Abstract

We prove that there exists a constant $c>0$ such that any finite group having no non-trivial mixed identity of length ≤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 ${\mathrm{PSL}}_n\left(q\right)$, ${\mathrm{PSp}}_{2m}\left(q\right)$, ${P\Omega }_{2m-1}^{\circ }\left(q\right)$, and ${\mathrm{PSU}}_n\left(q\right)$ 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.