Common transversals and complements in abelian groups

TL;DR

This paper investigates when multiple cyclic subgroups of a finite abelian group admit a common transversal or common complement, extending the classical two-subgroup case. It reduces existence questions to Sylow subgroups and employs the Krull–Remak–Schmidt framework to characterize complements, proving a general sufficient condition (Theorem A) for common complements of complemented isomorphic subgroups when the smallest prime dividing their order is large enough. For three cyclic subgroups of the same order, it provides a precise 2‑primary criterion, delineating when a common transversal or a common complement exists and outlining nonexistence cases; these results are complemented by explicit constructions. The work also delivers counting formulas for the number of common complements and offers constructive recipes that yield common transversals in several structured scenarios, including odd primes and homocyclic subgroups. Overall, the findings deepen understanding of transversal and complement structure in finite abelian groups and supply practical tools for explicit constructions.

Abstract

Given a finite abelian group $G$ and cyclic subgroups $A$, $B$, $C$ of $G$ of the same order, we find necessary and sufficient conditions for $A$, $B$, $C$ to admit a common transversal for the cosets they afford. For an arbitrary number of cyclic subgroups we give a sufficient criterion when there exists a common complement. Moreover, in several cases where a common transversal exists, we provide concrete constructions.

Theorems & Definitions (24)

• Theorem A
• Theorem B
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Corollary 2.5
• Lemma 3.1
• Corollary 3.2
• Corollary 3.3