A conjecture on partitions of groups

Abstract

We conjecture that every infinite group $G$ can be partitioned into countably many cells $G={\bigcup }_{n\in \omega }{A}_n$ such that $cov(A_n{A}_n^{-1})=|G|$ for each $n\in \omega$. Here $cov(A)=\min \{|X|:X\subseteq G,G=XA\}$. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.