On a characterization of locally finite groups in terms of linear cellular automata

TL;DR

This paper characterizes locally finite groups by showing that for such groups, every surjective linear cellular automaton with finite-dimensional alphabet is also injective, establishing a precise algebraic-automaton correspondence.

Contribution

It provides a new characterization of locally finite groups through the properties of linear cellular automata, linking group theory and automata theory.

Findings

Surjective linear cellular automata over locally finite groups are injective.

The characterization holds for automata over real or complex finite-dimensional alphabets.

Provides a necessary and sufficient condition for local finiteness in terms of cellular automata.

Abstract

We prove that a group $G$ is locally finite if and only if every surjective real (or complex) linear cellular automaton with finite-dimensional alphabet over $G$ is injective.