Extension of Automorphisms of Subgroups of Abelian and Polycyclic-by-Finite Groups

Abstract

We classify the abelian groups G, for which the following property holds: for every subgroup H, every ϕ ∈ Aut (H) has an extension ψ ∈ Aut (G). We also classify the infinite polycyclic-by-finite groups and the finite nilpotent 2′-groups having this property. Fuchs, Bertholf, Walls and Tomkinson did similar work for groups which have the property that homomorphisms of its subgroups extend to the whole group.

Extension of Automorphisms of Subgroups

“…In [1], these investigations were continued focusing on extensions of automorphism of subgroups. We recall the precise definition given to this.…”

“…So, a necessary condition for a finite group G to be of injective type is that |Aut(H)| divides |Aut(G)|, for every subgroup H of G. Clearly, the group ‫ޚ‬ is an abelian group of injective that, by the result of Fuchs, is not quasiinjective. In [1], the abelian groups of injective type have been described. It follows that a finite abelian group is of injective type if and only if it is quasi-injective.…”