A conjecture on partitions of groups

TL;DR

This paper proposes a conjecture that infinite groups can be partitioned into countably many subsets with large covering properties and confirms it for groups of regular cardinality and some Abelian groups.

Contribution

The paper introduces a new conjecture on partitions of infinite groups and proves it for groups of regular cardinality and certain Abelian groups.

Findings

Confirmed the conjecture for groups of regular cardinality.

Validated the conjecture for some Abelian groups of arbitrary cardinality.

Established conditions under which the partition property holds.

Abstract

We conjecture that every infinite group $G$ can be partitioned into countably many cells $G=\bigcup_{n\in\omega}A_n$ such that $cov(A_nA_n^{-1})=|G|$ 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.