Direct products and elementary equivalence of polycyclic-by-finite groups

TL;DR

This paper characterizes when polycyclic-by-finite groups are elementarily equivalent and explores how this relates to their decompositions into direct products of indecomposable groups.

Contribution

It provides an algebraic criterion for elementary equivalence of polycyclic-by-finite groups and examines its connection to their direct product decompositions.

Findings

Elementary equivalence of G and H is characterized algebraically.

Elementary equivalence of multiple direct products relates to factors' equivalence.

Open question: whether property 1 implies elementary equivalence for G and H.

Abstract

We give an algebraic characterization of elementary equivalence for polycyclic-by-finite groups. Using this characterization, we investigate the relations between their elementary equivalence and the elementary equivalence of the factors in their decompositions in direct products of indecomposable groups. In particular we prove that the elementary equivalence of two such groups G,H is equivalent to each of the following properties: 1)Gx...xG (k times G) and Hx...xG (k times H) are elementarily equivalent for a strictly positive integer k; 2)AxG and AxH are elementarily equivalent for two elementarily equivalent polycyclic-by-finite groups A,B. It is not presently known if 1) implies elementary equivalence for any groups G,H.