Centralisers of finite groups in locally finite simple groups

TL;DR

This paper constructs examples of finite simple groups with large representations having only scalar endomorphisms, showing that some automorphisms of certain infinite groups have trivial centralisers.

Contribution

It provides counterexamples to Hartley's question and demonstrates the existence of finite simple groups with trivial centralisers in automorphism groups of infinite simple groups.

Findings

Finite simple groups with arbitrarily large representations and scalar endomorphism rings.

Existence of finite simple groups with trivial centralisers in automorphisms of $SL( olinebreak ext{infinity}, ext{finite field})$.

The smallest example is $A_6$ with $q=9$.

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