Divisor graphs and symmetries of finite abelian groups

Abstract

The algebraic composition of an element [Formula: see text] of a commutative binary structure [Formula: see text] is modeled by its divisor (pseudo)graph [Formula: see text] (which can be regarded as a simple graph if [Formula: see text] is an abelian group), whose vertices are the elements of [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. With respect to the action of a group of symmetries on the set of divisor pseudographs of [Formula: see text], the numbers of orbits are considered, and these numbers are shown to yield groupoid-isotopy invariants when [Formula: see text] is a finite abelian group. The minimum number of orbits is computed for every abelian group of order at most 11, and it is also determined for all finite cyclic groups. Moreover, systems of pseudographs that can be realized as those of finite abelian groups are completely characterized. In fact, recurrence relations are given for constructing systems of divisor pseudographs of finite cyclic groups, and all commutative binary structures that are isotopic to a finite abelian group are established.