Centralisers of finite groups in locally finite simple groups

Abstract

We answer in the negative a question of Hartley about representations of finite groups, by constructing examples of finite simple groups with arbitrarily large representations whose endomorphism ring consists of just the scalars. We show as a consequence that there are finite simple groups of automorphisms of the locally finite simple group SL(∞, F q ) with trivial centraliser. The smallest of our examples is A 6 with q = 9.