Half-homomorphisms of groups

Abstract

Let G and G' be multiplicative systems. A half-homomorphism of G into G' will mean a mapping a---a' of G into G' such that for all a, bEG, (ab)'=a'b' or b'a'. An anti-homomorphism is a mapping such that always (ab)'=b'a'. The terms half-isomorphism, etc., are defined similarly. It will be shown that any half-homomorphism of a group G into a group G' is either a homomorphism or an anti-homomorphism (Theorem 2). The corresponding theorem for nonassociative rings (with the added requirement that (a+b)'=a'+b') was proved by Hua [1] (see also Jacobson and Rickart [2, Lemma 1]). For halfisomorphisms, it is sufficient to assume that G and G' are cancellation semigroups in order to obtain the analogous result (Theorem 1). Elxamples are given to show that Theorem 1 is false for semi-groups and for loops.