Infinite systems of equations in abelian and nilpotent groups
Group Theory
2026-01-13 v3
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.
Keywords
Cite
@article{arxiv.2410.20729,
title = {Infinite systems of equations in abelian and nilpotent groups},
author = {Mikhail A. Mikheenko},
journal= {arXiv preprint arXiv:2410.20729},
year = {2026}
}
Comments
12 pages; v2: some misprints corrected, added thanks to Anton Klyachko for the example from Question 1; v3: minor changes done, thanks to an anonymous referee