On a generalization of Shmel'kin's theorem

TL;DR

This paper investigates the conditions under which various classes of nilpotent groups, including periodic and torsion-free, contain solutions to all infinite unimodular systems of equations, extending known results from abelian groups.

Contribution

It generalizes Shmel'kin's theorem to broader classes of nilpotent groups, exploring solution existence for infinite systems of equations.

Findings

Periodic nilpotent groups contain solutions to all infinite unimodular systems.

Torsion-free nilpotent groups also contain solutions to all such systems.

The results extend the understanding of solution existence beyond abelian groups.

Abstract

It is known that every nilpotent group contains solution of every finite unimodular system of equatiuons over itself. This statement, however, is not true for infinite systems. Moreover, there are abelian groups which disprove the infinite system analogue of the statement. It has already been researched which periodic abelian groups contain solutions of all infinite unimodular systems of equations over themselves. The present article covers the same question for periodic nilpotent groups and for torsion-free nilpotent groups.