Hopfian combinatorial wreath products

TL;DR

This paper investigates conditions under which combinatorial wreath products of abelian groups are Hopfian, providing new examples and characterizations of automorphism groups in this context.

Contribution

It generalizes previous results on Hopfian properties of wreath products and offers new examples and automorphism group descriptions under specific conditions.

Findings

Identifies sufficient conditions for wreath products to be Hopfian.

Provides an example where the wreath product is not Hopfian despite the factors being Hopfian.

Describes automorphism groups of wreath products with certain restrictions.

Abstract

Let $A$ be an abelian group. We consider sufficient conditions for the combinatorial wreath product $A \wr_X B$ to be Hopfian generalising results of Bradford and Fournier-Facio. For an integer $m \geq 2$ we show an example where $\mathbb{Z}/ \mathbb{Z}_m \wr_X B$ is not Hopfian but $B$ is Hopfian. We describe $Aut(A \wr_X B)$ under some restrictions on $A$, $B$ and $X$.