On the functional graph of the power map over finite groups

TL;DR

This paper analyzes the structure of functional graphs generated by power maps over finite groups, providing new classifications for abelian and non-abelian groups, including flower groups like dihedral and quaternion groups.

Contribution

It offers a structural description of these graphs for abelian and flower groups, introducing new classifications and recursive descriptions for central trees.

Findings

Classifies isomorphism types of graphs for abelian groups

Identifies multiple classes of trees in flower groups

Provides recursive descriptions for central trees

Abstract

In this paper we study the description of the functional graphs associated with the power maps over finite groups. We present a structural result which describes the isomorphism class of these graphs for abelian groups and also for flower groups, which is a special class of non abelian groups introduced in this paper. Unlike the abelian case where all the trees associated with periodic points are isomorphic, in the case of flower groups we prove that several different classes of trees can occur. The class of central trees (i.e. associated with periodic points that are in the center of the group) are in general non-elementary and a recursive description is given in this work. Flower groups include many non abelian groups such as dihedral and generalized quaternion groups, and the projective general linear group of order two over a finite field. In particular, we provide improvements on past works regarding the description of the dynamics of the power map over these groups.