Twisted tensor product subgroups of finite classical groups

Abstract

If Cln(qr) denotes a classical group with natural module W of dimensioa n over Fqr , then the twisted tensor product module is realised over Fq, and yields an embedding . These embeddings play a significant role in the subgroup structure of classical groups; for example, Seitz [18] shows that any maximal absolutely irreducible subgroup defined over a proper extension field of Fq is of this form. In this paper we study the precise nature of these embeddings, and go on to investigate their maximality or otherwise. We show that the normaliser of Cln(qr) is usually maximal, with an explicit list of just 4 families of exceptions.