On the Fixity of Linear Groups

The fixity of a linear group G ⊆ GL( V ) is defined to be the maximal dimension of a centraliser C v ( g ) for g ∈ G with g ≠ 1. We study the structure of finite non‐modular linear groups of given fixity f . Our results may be regarded as an extension of the theory of Frobenius complements on the one hand, and of Jordan's classical theorem on finite subgroups of GL n (C) on the other. As an application we show that if G is a group of automorphisms of a finite group H , and each non‐trivial element of G fixes at most f points of H , then G has a soluble subgroup of derived length at most 3 whose index is bounded above in terms of f alone.