Common transversals and complements in abelian groups

TL;DR

This paper investigates conditions under which cyclic subgroups of finite abelian groups admit common transversals and complements, providing necessary and sufficient criteria along with explicit constructions in certain cases.

Contribution

It establishes new necessary and sufficient conditions for common transversals and complements among cyclic subgroups in finite abelian groups, including explicit construction methods.

Findings

Necessary and sufficient conditions for common transversals of three cyclic subgroups.

A sufficient criterion for the existence of common complements among multiple cyclic subgroups.

Concrete constructions of common transversals in several cases.

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.