ZS-metacyclic groups and their endomorphism near-rings
The first author and \textit{C. G. Lyons} started the study of endomorphism near-rings [in Proc. Edinb. Math. Soc., II. Ser. 17, 71-78 (1970; Zbl 0203.336)]. This paper follows in that line and determines the endomorphism near-rings of a large class of groups, using an extension of the methods developed in the paper above. Let \(S\) be a semigroup of endomorphisms of the group \(G\). Then \(S\) generates a near-ring of mappings from \(G\) to itself. The three important cases are \(S = \text{Inn }G\), the inner automorphisms, \(S = \text{Aut }G\), all the automorphisms, \(S = \text{End }G\), all the endomorphisms. The corresponding nearrings are called \(I(G)\), \(A(G)\) and \(E(G)\) respectively. The groups considered here are ZS-metacyclic groups, which are extensions of a cyclic group by another cyclic group of order prime to the first. This is equivalent to saying that all the Sylow subgroups are cyclic. The authors determine all the endomorphisms of a ZS-metacyclic group \(G\). They use a Peirce decomposition method to obtain the structure of \(I(G)\). Then they show that \(I(G) = E(G)\), so every endomorphism of \(G\) is a sum of inner automorphisms. This is a major advance in the study of \(I\)-\(E\) groups, groups \(H\) for which \(I(H) = E(H)\). Finally the radical of \(I(G)\) is determined. As may be expected, there is a great deal of careful detailed work in this paper.
scientific article; zbMATH DE number 729092
scientific article