E-groups do not have proper semidirect sum decompositions

Abstract

An E-group is a group in which each element commutes with any of its endomorphic images. Hence E-groups provide a generalization of abelian groups. It is easy to show that the endomorphism near-ring (the group generated additively by the endomorphisms) of an E-group is a ring, just as the endomorphism near-ring of an abelian group is a ring. In this paper, we establish the fact that, like abelian groups, E-groups have no proper semidirect sum decompositions (i.e. a semidirect sum decomposition of an E-group must be a direct sum decomposition).

Direct product of four classes of groups

“…Since G is a nonabelian E-group, G = H × K by Boudreaux (2007b). Now H is nonabelian and it is a homomorphic image of G, hence H is a nonabelian E-group.…”