Digraphs of power maps over finite nilpotent groups

Abstract

Given a finite group G and an integer t, let G(φt/G) be the directed graph with vertex set G and directed edges g→gt,g∈G. We prove that nilpotent groups are precisely the class of groups yielding a special symmetry on the graphs G(φt/G) that has appeared several times in the literature (considering functions over various algebraic structures). We then explore the question on when two finite nilpotent groups yield the same digraphs as t runs over Z: this defines an equivalence relation ∼φ. We completely describe the set of integers M in which the relation ∼φ is equivalent to the isomorphism of groups when one restricts to the set of nilpotent groups of order M. Our proofs rely on arguments from number theory and combinatorics.