ZS-metacyclic groups and their endomorphism near-rings

Abstract

Abstract. If G is a group written additively, the inner automorphisms and the endomorphisms additively generate near-rings

Semidirect products of 𝐼-𝐸 groups

“…We cannot always have such a semidirect product being an I-E group since dihedral groups of order 2/j are not 7-is when n is even by [3], but what about when the factors have relatively prime orders? In the next section, we shall show that if G is the semidirect product of a normal 7-E group H and an I-E group K with (\H\, \K\) = I, Xhe problem of determining when G is I-E reduces to considering whether one endomorphism lies in 1(G), namely, the projection of G onto K. We then use this result in the third and final section of this paper to prove that the semidirect product of two cyclic groups of relatively prime orders is I-E, thereby eliminating the assumption in [4] that the normal cyclic subgroup is the commutator subgroup. However, after we complete this proof, we shall point out that the cyclic subgroups may be reselected so that they still have relatively prime orders and the normal one is the commutator subgroup.…”

“…From [2] we see that dihedral groups of order 2« are 7-E when n is odd. Recently, Malone and Mason have generalized this dihedral result in [4] by showing that a semidirect product of cyclic groups of relatively prime orders is I-E when the cyclic normal subgroup is the commutator subgroup. In [5,Theorem 4.11], it is shown that the symmetric groups S" are 7-E groups when n > 5.…”

“…Noting the groups in [4] and [5] are semidirect products of I-E groups, it is natural to ask whether there are any general conditions under which a semidirect product of I-E groups is an 7-E group. We cannot always have such a semidirect product being an I-E group since dihedral groups of order 2/j are not 7-is when n is even by [3], but what about when the factors have relatively prime orders?…”

The semidirect products of finite cyclic groups that areI-E groups

“…Suppose that G is the semidirect product of a finite cyclic normal subgroup A and a finite cyclic group B. In [6], MALONE and MASON showed that G is an I-E group when (IAJ, ]BI)= 1 and A = G'. In [5,Theorem 3.2], C. G. LYONS and this author showed that the assumption A = G' can be dropped so that any semidirect product of cyclic groups of relatively prime orders is an I-E group.…”

Semidirect sum of groups in which endomorphisms are generated by inner automorphisms

“…Recently, questions concerning the structure of I-E groups have been considered. J. J. Malone and G. Mason [17] Their results depend heavily on [10, Theorem 2.1], which characterizes when a semidirect product of I-E groups of relatively prime orders is an I-E group. In this paper, not assuming G to be finite, this characterization theorem is generalized and, at the same time, a fairly concise proof is provided in Theorem 2.1.…”