On the dynamics of endomorphisms of finite groups

TL;DR

This paper studies the structure of finite groups as dynamical systems under endomorphisms, characterizing their state spaces and extending previous results to cyclic groups of arbitrary order.

Contribution

It generalizes existing results on linear finite dynamical systems to all finite groups, providing a complete characterization of state space structures and counting isomorphism types.

Findings

State spaces are graph tensor products of directed 1-trees and cycles.

Full characterization of nilpotent endomorphism state spaces.

Count of isomorphism types for endomorphisms of finite cyclic groups.

Abstract

Aiming at a better understanding of finite groups as finite dynamical systems, we show that by a version of Fitting's Lemma for groups, each state space of an endomorphism of a finite group is a graph tensor product of a finite directed $1$-tree whose cycle is a loop with a disjoint union of cycles, generalizing results of Hern\'andez-Toledo on linear finite dynamical systems, and we fully characterize the possible forms of state spaces of nilpotent endomorphisms via their "ramification behavior". Finally, as an application, we will count the isomorphism types of state spaces of endomorphisms of finite cyclic groups in general, extending results of Hern\'andez-Toledo on primary cyclic groups of odd order.