On kernels of cellular covers

Abstract

In the present paper we continue to examine cellular covers of groups, focusing on the cardinality and the structure of the kernel $K$ of the cellular map $G\to M$. We show that in general a torsion free reduced abelian group $M$ may have a proper class of non-isomorphic cellular covers. In other words, the cardinality of the kernels is unbounded. In the opposite direction we show that if the kernel of a cellular cover of any group $M$ has certain “freeness” properties, then its cardinality is bounded by $|M|$.