A non-abelian 2-group whose endomorphisms generate a ring, and other examples of E-groups
Groups for which the distributively generated near-ring generated by the endomorphisms is in fact a ring are known as E-groups and are discussed in (3). R. Faudree in (1) has given the only published examples of non-abelian E-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.
Endomorphism rings of non-abelian groups
In 1942 F.W. Levi described those groups in which any two inner automorphic images of an arbitrary element commute. Until recently it was not known whether there existed non-abelian groups with the property that any two endomorphic images of an arbitrary element commute. Now, R. Faudree has given examples of finite p–groups having this property. In this paper we give a necessary and sufficient condition that a torsion group which contains no elements of order 2 has this property. The technique of proof involves looking at certain near rings of functions on the group. The inspiration for the theorem comes from the “halving” technique used by B.H. Neumann in 1940.
Endomorphism near-rings
The study of near-rings is motivated by consideration of the system generated by the endomorphisms of a (not necessarily commutative) group. Such endomorphism near-rings also furnish the motivation for the concept of a distributively generated (d.g.) near-ring. Although d.g. near-rings have been extensively studied, little is known about the structure of endomorphism near-rings. In this paper results are presented which enable one to give the elements of the endomorphism near-ring of a given group. Also, some results relating to the right ideal structure of an endomorphism near-ring are presented. These concepts are applied to present a detailed picture of the properties of the endomorphism near-ring of (S3, +).