Twisted tensor product subgroups of finite classical groups

Abstract

If Cl_n(q^r) denotes a classical group with natural module W of dimensioa n over F_q^r , then the twisted tensor product module is realised over F_q, and yields an embedding

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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 F_q 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 Cl_n(q^r) is usually maximal, with an explicit list of just 4 families of exceptions.