Asymmetrically decomposable Abelian groups (Q1358509)

An infinite Abelian group \(G\) is said to be asymmetrically decomposable if \(G\) can be decomposed into two subsets \(A_1\) and \(A_2\) such that \((g+S)\cap A_1\neq\emptyset\) and \((g+S)\cap A_2\neq\emptyset\) for every element \(g\in G\) and any infinite symmetric subset \(S\subseteq G\). Recall that a subset \(S\) is symmetric if \(S=-S\). The aim of the present note is to prove the following Theorem. A group \(G\) is asymmetrically decomposable if and only if \(G\) belongs to one of the following two classes of groups: (1) countable periodic groups with a finite number of elements of order 2; (2) groups of the form \(\mathbb{Z}\times K\), where \(K\) is a finite group.