Hall–Higman‐type theorems for semisimple elements of finite classical groups

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.