Cyclic-quasi-injective for Finite Abelian Groups

TL;DR

We address the problem of characterizing finite abelian groups $G$ for which every cyclic subgroup $H$ and every homomorphism $f:H\to G$ extends to an endomorphism of $G$ (cyclic-quasi-injectivity). The authors develop a $p$-primary framework, introduce obstruction sets $X(G)$ and $Y(G)$, and prove a bijection between $X(G)/\sim$ and $Y(G)$, yielding necessary and sufficient conditions in terms of $p$-components being homocyclic. They provide explicit counting formulas for $|X(G)|$ and $|X(G)/\sim|$ via detailed combinatorial expressions and establish a decomposition principle across coprime components to characterize when $G$ is cyclic-quasi-injective. An application connects $|X(G)/\sim|$ to permutation-jump statistics, giving exact identities such as $|X(G)/\sim|=\sum_{\sigma\in S_n}\max_i(\sigma(i)-i)$ for certain $G$, thereby linking the algebraic structure to combinatorial invariants. The results advance understanding of how endomorphism extendability constraints interact with the $p$-adic structure of finite abelian groups and reveal a rich interplay between group theory and permutation combinatorics.

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 $\sum_{σ\in S_{n}} \max_{1 \leq i \leq n} \{σ(i) - i\}$.

Theorems & Definitions (27)

• Theorem 1.1
• Example 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4