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Mikhail A. Mikheenko

Publications and source records attributed to Mikhail A. Mikheenko.

7 recordsLinked to original sources
arXiv2026

Solvability of unimodular equations in groups and Lie algebras

Our results implies, in particular, that a finitely generated solvable group $G$ is nilpotent if and only if it contains a solution to any unimodular equation, i.e., an equation of the form $\prod g_ix^{n_i}=1$, where $g_i\in G$ and $\sum n_i=\pm1$. A similar fact turns out to be true for Lie algebras. We also exhibit an example of a unimodular equation $w(x)=g$ over a finitely generated group $G$, which has a solution (in $G$) for any $g\in G$, but the solution is not unique for some $g\in G$. We show that, for nilpotent groups $G$, the set of unimodular mappings $G^n\to G^n$ (which are defined naturally) forms a group under the composition.

math.GR↗
arXiv2026

On a generalization of Shmel'kin's theorem

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.

math.GR↗
arXiv2025

Infinite systems of equations in abelian and nilpotent groups

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.

math.GR↗
arXiv2025

Unimodular equations which do not preserve the derived length of a group

It is a known fact that any unimodular equation over an abelian group has a solution in that group itself. It is also known that for metabelian groups this does not hold; moreover, there is a unimodular equation over some metabelian group which has no solutions in any larger metabelian group. Here we present the proof of an analagous fact for solvable groups of higher derived lengths.

math.GR↗
arXiv2024

On $p$-nonsingular systems of equations over solvable groups

Any group that has a subnormal series, in which all factors are abelian and all except the last one are $p'$-torsion-free, can be embedded into a group with a subnormal series of the same length, with the same properties and such that any $p$-nonsingular system of equations over this group is solvable in this group itself. This helps us to prove that the minimal order of a metabelian group, over which there is a unimodular equation that is unsolvable in metabelian groups, is 42.

math.GR↗
arXiv2023

Equations over solvable groups

Not any nonsingular equation over a metabelian group has solution in a larger metabelian group. However, any nonsingular equation over a solvable group with a subnormal series with abelian torsion-free quotients has a solution in a larger group with a similar subnormal series of the same length (and an analogous fact is valid for systems of equations).

math.GR↗