Multiplication Groups of Abelian Torsion-Free Groups of Finite Rank

TL;DR

This paper characterizes the multiplication groups of a specific class of Abelian torsion-free groups of finite rank, showing they belong to the same class and describing their structural invariants.

Contribution

It provides a detailed description of the multiplication groups for reduced block-rigid almost completely decomposable Abelian groups of ring type with cyclic regulator quotient, establishing their structural properties.

Findings

Multiplication groups of G are also in class A_0.

Explicit descriptions of rank, regulator, and invariants of Mult G.

Mult G admits a main decomposition and standard representation.

Abstract

For an Abelian group $G$, any homomorphism $\mu\colon G\otimes G\rightarrow G$ 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 $\text{Mult}\,G$.