The length of mixed identities for finite groups

Abstract: We prove that there exists a constant $c&gt;0$ 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 ${\rm PSL}<em>n(q)$, ${\rm PSp}</em>{2m}(q)$, ${\rm P \Omega}_{2m-1}<sup>\circ(q)$, 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 $q$.