A non-abelian 2-group whose endomorphisms generate a ring, and other examples of E-groups

Abstract

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.

An interesting orbit-induced partition of the class of all groups

“…This is an infinite family of finite p-groups. It was Malone [14] who pointed out that p must be odd for Faudree's construction to work.…”

“…All of their groups have the property that (Aut(G), •) is abelian. Malone [14] designates the ones that are E-D groups as JK groups, and notes that, like the F p 8 groups, the JK groups have order p 8 . Since p can be any prime natural number, this construction yielded the first published example of an E-D group that is a 2-group.…”

Finite p-groups

“…Examples of pE'groups are constructed in [M:340, KMC:1042, [183], KMC:566]. It is well known (Huppert [ChSh:472, Sh 6.4-6,5]) (Levi (1942)) that the nilpotency length of an E-group is not higher than 3, and if that group does not have elements of order 3, its nilpotency length is not even higher than 2.…”

The group of automorphisms of a distributively generated near ring

“…Let Qn designate the generalized quaternion group of order 2", n > 3. (See [3] for a discussion of the facts about Qn which are cited below). In Qn an element of order 4 not contained in the cyclic subgroup of order 2"~] can be mapped under an appropriate automorphism to any element of order 4 not in the cyclic subgroup of order 2"~ '.…”