A property of finite simple non-abelian groups

Abstract

It has been proved by Frohlich [1] that the collection of inner automorphisms of a finite simple non-abelian group G generates the group (under pointwise multiplication) of all functions from G into G leaving the identity fixed, and, conversely, that the only finite groups with this property are the simple non-abelian groups, Z2, and [e]. It follows directly that the inner automorphisms and the constant functions of a simple non-abelian group into itself generate the entire group of functions from G into G. The purpose of this note is to prove the following generalization of this theorem.