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On a generalization of Shmel'kin's theorem

Mikhail A. Mikheenko

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.

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.

Paper Structure

This paper contains 5 sections, 20 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Divisible subgroups of nilpotent groups
  3. Periodic nilpotent groups
  4. Torsion-free groups
  5. Open questions

Key Result

Theorem 1

Any nonsingular system of equations over a finite group is solvable over this group.

Theorems & Definitions (37)

  • Definition
  • Definition
  • Theorem : GR62
  • Theorem : How81
  • Theorem : G83, see also Kr85
  • Theorem : M25
  • Theorem : M25
  • Theorem : M25
  • Theorem 1
  • Theorem 2
  • ...and 27 more