Hopfian combinatorial wreath products

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 $m\ge 2$ 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 $X$.