Multiplication Groups of Abelian Torsion-Free Groups of Finite Rank

Abstract

For an Abelian group $G$, any homomorphism $\mu :G\otimes G\to G$ is called a \textsf{multiplication} on $G$. The set $\text{Mult}G$ of all multiplications on an Abelian group $G$ itself is an Abelian group with respect to addition; the group is called the \textsf{multiplication group} of $G$. Let ${\mathcal{A}}_0$ be the class of all reduced block-rigid almost completely decomposable groups of ring type with cyclic regulator quotient. In this paper, for groups $G\in {\mathcal{A}}_0$, we describe groups $\text{Mult}G$. We prove that for $G\in {\mathcal{A}}_0$, the group $\text{Mult}G$ also belongs to the class ${\mathcal{A}}_0$. For any group $G\in {\mathcal{A}}_0$, we describe the rank, the regulator, the regulator index, invariants of near-isomorphism, a main decomposition, and a standard representation of the group $\text{Mult}G$.