Complete groups with nonabelian composition factors

Abstract:

A finite group is said to be complete if it has trivial center and if every automorphism is an inner automorphism. A finite group with nonabelian composition factors has a unique completely reducible radical (CR radical). We consider finite groups with nonabelian composition factors whose CR radical consists of complete simple groups and we give necessary and sufficient conditions for such a group to be complete. This involves finding group theoretic conditions which are necessary and sufficient for a finite centerless group to occur as a self-normalizing subgroup of a direct product of symmetric groups.