A note on extending partial automorphisms of abelian groups

Abstract

Given a group G and a partial automorphism μ of G, i.e. an isomorphism mapping a subgroup A of G onto another subgroup B of G, then it is known [3] that μ can always be extended to a total automorphism, in fact an inner one, of a supergroup of G; that is there exists a group G* ⊇ G with an inner automorphism μ* whose effect on the elements of A is the same as that of μ. Also any number of partial automorphisms μσ, where a ranges over some index set Σ can be simultaneously extended to inner automorphisms of one and the same group [3, Theorem II].