Factors in infinite groups

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.