  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 $SL(\infty,\mathbb{F}_q)$ with
trivial centraliser. The smallest of our examples is $A_6$ with $q=9$.
