Minimal generating sets in wreath products

TL;DR

This paper investigates conditions under which wreath products of groups have minimal generating sets, demonstrating this for automorphism groups of regular rooted trees and their finite-state automorphisms.

Contribution

It provides new sufficient conditions for wreath products to have minimal generating sets, with specific results for automorphism groups of regular rooted trees.

Findings

Wreath products can have minimal generating sets under certain conditions.

Automorphism groups of regular rooted trees satisfy these conditions.

Finite-state automorphisms of such trees also have minimal generating sets.

Abstract

We find some sufficient conditions under which the permutational wreath product of two groups has a minimal generating set. In particular we prove that for a regular rooted tree the group of all automorphisms and the group of all finite-state automorphisms of such a tree satisfy these conditions.