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Hopfian combinatorial wreath products

Dessislava H. Kochloukova

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 $X$.

Hopfian combinatorial wreath products

Abstract

Let be an abelian group. We consider sufficient conditions for the combinatorial wreath product to be Hopfian generalising results of Bradford and Fournier-Facio. For an integer we show an example where is not Hopfian but is Hopfian. We describe under some restrictions on , and .
Paper Structure (4 sections, 11 theorems, 46 equations)

This paper contains 4 sections, 11 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Hopfian and non-Hopfian wreath products
  3. Applications of Corollary D
  4. The automorphism group of $G = A \wr_X B$

Key Result

Theorem 2.1

Lu Let $B$ acts on $X$ with finitely many orbits, $B(x) = stab_B(x)$ and for every two $x,y \in X$ assume that Then for a field $K$ of characteristic zero the ring $End_{K B}( K X)$ is directly finite.

Theorems & Definitions (21)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 11 more