Infinite systems of equations in abelian and nilpotent groups

TL;DR

The paper investigates conditions under which infinite systems of equations have solutions in abelian and nilpotent groups, establishing criteria for solvability in various classes of these groups.

Contribution

It introduces a new criterion for periodic abelian groups to solve infinite unimodular systems and extends solvability results to nilpotent groups of bounded period and divisible nilpotent groups.

Findings

Periodic abelian groups solve all infinite unimodular systems under the new criterion.

Nilpotent groups of bounded period also solve all such systems.

Divisible nilpotent groups solve all nonsingular infinite systems.

Abstract

Every abelian (and even every nilpotent) group contains a solution of any finite unimodular system of equations over itself. However, this is not true for infinite systems. We deduced a criterion for a periodic abelian group to contain a solution of any infinite unimodular system of equations over itself. Using this criterion, we show that nilpotent groups of bounded period also contain solutions of all infinite unimodular systems of equations over themselves. Solvability of every nonsingular infinite system of equations in each divisible nilpotent group is shown as well.