An analogue to Dixon's theorem for automaton groups

TL;DR

This paper explores the distribution of finite groups generated by a special class of Mealy automata, revealing that the groups are typically structured as semi-direct products involving alternating groups and transpositions, contrasting with uniform randomness.

Contribution

It introduces an analogue to Dixon's theorem for automaton groups, analyzing the typical structure of groups generated by finite automata.

Findings

Generated groups are usually semi-direct products of alternating groups and transposition groups.

Distribution of generated groups is far from uniform, showing a bias towards specific structures.

Automaton-generated groups do not follow a uniform distribution, unlike random permutations.

Abstract

Dixon's famous theorem states that the group generated by two random permutations of a finite set is generically either the whole symmetric group or the alternating group. In the context of random generation of finite groups this means that it is hopeless to wish for a uniform distribution -- or even a non-trivial one -- by drawing random permutations and looking at the generated group. Mealy automata are a powerful tool to generate groups, including all finite groups and many interesting infinite ones, whence the idea of generating random finite groups by drawing random Mealy automata. In this paper we show that, for a special class of Mealy automata that generate only finite groups, the distribution is far from being uniform since the obtained groups are generically a semi-direct product between a direct product of alternating groups and a group generated by a tuple of transpositions.