Infinite systems of equations in abelian and nilpotent groups

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.