# Automata system in finitelly generated groups

## TL;DR

This paper investigates the capabilities of finite automata systems in exploring Cayley graphs of groups, establishing limitations based on the group's periodicity and providing conditions under which exploration is possible.

## Contribution

It proves that finite automata systems cannot leave certain areas in Cayley graphs of periodic groups and characterizes exploration possibilities for non-periodic and aperiodic groups.

## Key findings

- Finite automata cannot explore some regions of Cayley graphs of periodic groups.
- Groups with non-periodic elements can be explored by finite automata with 3 pebbles.
- Finitely generated aperiodic groups cannot be explored by any finite automata system.

## Abstract

We prove that any finite system of interacted automata can not leave some finite arear of Calley graph of periodic group. If group has non-periodic element, then its Calley graph can be explored by some finite automata with 3 pebbles. If group is finitely generated and aperiodic then it can not be explored by any system of finite automata.