A Non-Abelian 2-Group Whose Endomorphisms Generate a Ring, and other Examples of E-Groups

1. Introduction Groups for which the distributively generated near-ring generated by the endomorphisms is in fact a ring are known as U-groups and are discussed in (3). R. Faudree in (1) has given the only published examples of non-abelian JS-groups by presenting defining relations for a family of p-groups. However, as shown in (3), Faudree's group does not have the desired property when p = 2. In this note, it is shown that most of the groups discussed by D. Jonah and M. Konvisser in (2) are actually E-groups. These groups, described below in Section 2 are proved by Jonah and Konvisser to be such that all their automorphisms are central. Here, it is shown that most of these groups are E-groups by proving that each strict endomorphism (i.e. an endomorphism that is not an automorphism) has its image in the centre of the group. Since one of the groups treated is a 2-group, this paper provides the only published example of a non-abelian 2-group which is an E-group. E-groups are also discussed in (4) and (5). However, no examples are given in those papers.