Extension of partial endomorphisms of abelian groups

Extract

It is known [1] that for a partial endomorphism μ of a group G that maps the subgroup A ⊆ G onto B ⊆ G. G to be extendable to a total endomorphism μ^* of a supergroup G^* ⊆ G such that μ an isomorphism on G*(μ^*)^m for some positive integer m, it is necessary and sufficient that there exist in G a sequence of normal subgroups

such that L₁ ƞA is the kernel of μ and

for ι = 1, 2,…, m–1.

The question then arises whether these conditions could be simplified when the group G is abelian. In this paper it is shown not only that the conditions are simplified when Gis abelian but also that the extension group G^*⊇G can be chosen as an abelian group.