Divisor graphs and symmetries of finite abelian groups

The algebraic composition of an element x of a commutative binary structure B is modeled by its divisor (pseudo)graph Gamma(x) (which can be regarded as a simple graph if B is an abelian group), whose vertices are the elements of B such that two vertices a and b are adjacent if and only if ab = x. With respect to the action of a group of symmetries on the set of divisor pseudographs of B, the numbers of orbits are considered, and these numbers are shown to yield groupoid-isotopy invariants when B 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.