Cyclic-quasi-injective for Finite Abelian Groups

TL;DR

This paper characterizes finite abelian groups where homomorphisms from cyclic subgroups can be extended to the whole group, providing necessary and sufficient conditions and linking subgroup properties to permutation jumps.

Contribution

It establishes criteria for cyclic-quasi-injective groups and quantifies the non-extendable homomorphisms using permutation analysis.

Findings

Identifies necessary and sufficient conditions for cyclic-quasi-injectivity.

Calculates the number of cyclic subgroups with non-extendable homomorphisms.

Connects subgroup extension properties to permutation maximum jumps.

Abstract

We investigate the conditions for a finite abelian group $G$ under which any cyclic subgroup $H$ and any group homomorphism $f \in \operatorname{Hom}(H,G)$ can be extended to an endomorphism $F \in \operatorname{End}(G)$. As a result, we provide necessary and sufficient conditions for such a group $G$ and we compute the number of cyclic subgroups possessing non-extendable homomorphisms. In addition, we demonstrate that the number of cyclic subgroups that do not satisfy the conditions corresponds to the sum of the maximum jumps in the associated permutations given by $\sum_{\sigma \in S_{n}} \max_{1 \leq i \leq n} \{\sigma(i) - i\}$.