Group homomorphism generated near-rings and rings: A unity not fixing each element of the group

摘要

Let (G, +) be a group, not necessarily abelian, and let K be a nontrivial subgroup of G. Let H = A(G, K) be the additive group generated by Hom (G, K). Then (H(G, K), +, o) is a d.g. near-ring. If K not equal G, then H(G, K) cannot contain the unity element of E(G), the near-ring generated by End G. Surprisingly, examples exist which show it may indeed have a two-sided unity element. Conditions are developed involving G and (K) over bar, the additive subgroup generated by U {h(G): h is an element of H}, which characterize when H(G, K) contains a one-sided or two-sided unity element. The cases when (K) over bar is abelian or an E-group are considered. As a consequence of this theory, connections between E(G) and E((K) over bar), via H(G, K), are established. Numerous illustrative examples are given. [References: 5]