Embedding Theorems for Groups

By a partial endomorphism of a group G we mean a homomorphic mapping μ of a subgroup A of G onto a subgroup B of G. If μ is denned on the whole of G then it is called a total endomorphism. We call a partial endomorphism totally extendable (or extendable) if there exists a supergroup G*⊇G with a total endomorphism μ* which extends μ in the sense that gμ* = gμ, whenever the right-hand side is defined (3).