Structure of the invertible CA transformations group

Abstract

We describe the structure of the group of all invertible CA transformations acting on 1-dimensional finite-length cellular automata defined on a finite states set. It turns out that the group is a direct product of semidirect products of cyclic and symmetric groups. The analysis of this group has been carried out by means of an isomorphic image of the invertible CA transformations group, which was easier to handle. A presentation of the group by generators and relations is also supplied. Most of the results obtained can also be applied to analyze the automorphism group of any finite one-to-one dynamical system.