Hall-Higman-type theorems for semisimple elements of finite classical groups

Abstract

Minimum polynomials of semisimple elements of prime power order p a of finite classical groups in (nontrivial) irreducible cross-characteristic representations are studied. In particular, an analogue of the Hall-Higman theorem is established, which shows that the degree of such a polynomial is at least p a−1 (p−1), with a few explicit exceptions.