Subgroups which admit extensions of homomorphisms

TL;DR

This paper classifies finite subgroups of primary abelian groups based on their extension properties of homomorphisms and applies these classifications to finitely generated subgroups, revealing that monomorphism extendibility implies all homomorphisms extend.

Contribution

It provides a numerical invariant classification of subgroups with extension properties and links monomorphism extendibility to all homomorphisms for finitely generated subgroups.

Findings

Classified finite subgroups by extension properties using numerical invariants.

Showed that for finitely generated subgroups, monomorphism extendibility implies all homomorphisms extend.

Established criteria for when homomorphisms extend from subgroups to entire groups.

Abstract

We classify by numerical invariants the finite subgroups $H$ of a primary abelian group $G$ for which every homomorphism or monomorphism of $H$ into $G$, or every endomorphism of $H$, extends to an endomorphism of $G$. We apply these results to show that for finitely generated subgroups of general abelian groups, the extendibility of monomorphisms implies the extendibility of all homomorphisms.