On a generalization of Shmel'kin's theorem

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.

20 pages; v2: the proof of Proposition 1 in Section 2 rewritten (thanks to A. Yu. Olshanskii), the torsion-free case completely rewritten (the previously stated result was false) in two sections instead of one