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Factors in infinite groups

Mikhail Kabenyuk

Abstract

Let $G$ be a group and $A\subseteq G$ a non-empty subset. A right $s$-factor associated with $A$ is a maximal subset $U\subseteq G$ such that the product $AU$ is direct. The lower and upper $s$-indices $|G:A|^-$ and $|G:A|^+$ are defined as the minimum and the supremum of the cardinalities of such maximal sets $U$. The subset $A$ is called stable if $|G:A|^- = |G:A|^+$, and $G$ is called stable if every subset of $G$ is stable. Using a graph-theoretic reformulation in terms of Cayley graphs, we prove that every infinite group is unstable. Equivalently, for every infinite group $G$ there exists a subset $A\subseteq G$ for which maximal subsets $U$ with direct product $AU$ do not all have the same cardinality. This gives a negative answer to Question 21.58 of the Kourovka Notebook.

Factors in infinite groups

Abstract

Let be a group and a non-empty subset. A right -factor associated with is a maximal subset such that the product is direct. The lower and upper -indices and are defined as the minimum and the supremum of the cardinalities of such maximal sets . The subset is called stable if , and is called stable if every subset of is stable. Using a graph-theoretic reformulation in terms of Cayley graphs, we prove that every infinite group is unstable. Equivalently, for every infinite group there exists a subset for which maximal subsets with direct product do not all have the same cardinality. This gives a negative answer to Question 21.58 of the Kourovka Notebook.
Paper Structure (6 sections, 8 theorems, 29 equations)

This paper contains 6 sections, 8 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Graph-theoretic reformulation
  3. A lemma on finite symmetric subsets
  4. Finite configurations forcing instability
  5. Auxiliary lemmas on 2-groups
  6. Proof of the theorem

Key Result

Lemma 1

Let $A\subset G$, let $F=G\setminus A^{-1}A$, and let $\Gamma=\mathop{\mathrm{Cay}}\nolimits(G,\partial A)$. Assume that $F$ is finite and $\Delta=\Delta(F)$. Then $\omega(\Delta)=\alpha(\Gamma)-1$, and $\iota(\Delta)=i(\Gamma)-1$. Consequently, the graph $\Gamma$ is well-covered if and only if all

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 6 more