On the dynamics of endomorphisms of finite groups

Alexander Bors

doi:10.1007/s00200-016-0304-9

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 Hernández-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 Hernández-Toledo on primary cyclic groups of odd order.

Highlights

Finite dynamical systems have recently gained a lot of interest within mathematics, and for their practical applications in areas such as cryptography, pseudorandom number generation and reverse engineering
There is a well-established theory of so-called linear finite dynamical systems
We generalize the results on LFDSs, but going in a different direction than usually: What if not f is replaced by a more complicated polynomial map, but we keep the “nice” property of f being an endomorphism and instead replace the vector space structure on the underlying set by a group structure? It turns out that the basic results on LFDSs mentioned in the last paragraph can be transferred to this more general situation

Summary

Some background

Finite dynamical systems have recently gained a lot of interest within mathematics, and for their practical applications in areas such as cryptography, pseudorandom number generation and reverse engineering. There is a well-established theory of so-called linear finite dynamical systems (a special case, abbreviated by LFDSs). These consist of a finitedimensional vector space V over a finite field together with a linear map f : V → V. The results give strong restrictions on the possible forms of state spaces compared to arbitrary finite dynamical systems. As we will see, the group laws impose strong restrictions on the form of the 1-tree representing the nilpotent part

Results on the structure of the state space

An application to finite cyclic groups