Subgroups which admit extensions of homomorphisms

Abstract

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.

Strictly Invariant Submodules

“…In this context we mention that in the case of Abelian groups, all pseudoinjective groups are quasi-injective, [21]. A similar situation occurred in [4]: denoting by Q(G), the family of all subgroups N ≤ G such that every homomorphism N −→ G extends to an endomorphism of G and by P(G), the family of all subgroups N ≤ G such that every injective homomorphism N −→ G extends to an endomorphism of G, though we strongly suspect that Q(G) = P(G) for Abelian groups, the proof which shows that finitely generated subgroups from P(G) are also in Q(G) was already very hard (and the general question is still open).…”

“…Injective invariant subgroups of Abelian groups were termed S-characteristic and left invariant, respectively, in [2] or [16]. These were used in [4] for the study of (co)hopfian modules.…”

Strongly inert subgroups of abelian groups

“…A straightforward verification shows that if K ≤ H, H is a direct summand of a group G and K is fully inert in G, then K is fully inert in H. Notice that the direct summand property may be weakened requiring K ∈ W(H), if W(H) denotes the class of all subgroups K of H such that all endomorphisms of K can be extended to endomorphisms of H (see [3]). The "strongly" analogue is immediate.…”

Additive unit structure of endomorphism rings and invariance of modules

“…In[3] it was asked whether in the case of abelian groups, the notions of automorphism-invariance and quasi-injectivity coincide. Our Theorem 4.11 answers this open question in the affirmative.…”