相关论文: Infinite systems of equations in abelian and nilpo…
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…
We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also…
Over each nontrivial finite group $G$, there exists a finite system of equations having no solutions in larger finite groups but having a solution in a periodic group containing $G$. We prove several similar facts about amenable, orderable,…
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…
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…
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…
In this paper we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more…
Recently, M. Kompatscher proved that for each finite supernilpotent algebra $\mathbf{A}$ in a congruence modular variety, there is a polynomial time algorithm to solve polynomial equations over this algebra. Let $\mu$ be the maximal arity…
The complexity of the equation solvability problem is known for nilpotent groups, for not solvable groups and for some semidirect products of Abelian groups. We provide a new polynomial time algorithm for deciding the equation solvability…
We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense,…
The Equation Problem in finitely presented groups asks if there exists an algorithm which determines in finite amount of time whether any given equation system has a solution or not. We show that the Equation Problem in central extensions…
Among other things, we prove that the group of automorphisms fixing every normal subgroup of a nilpotent-by-abelian group is nilpotent-by-metabelian. In particular, the group of automorphisms fixing every normal subgroup of a metabelian…
We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example: Theorem: For $n\ge 2$, there is a…
In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.
In this paper, we show how solutions to explicit algebraic systems lead to solutions to infinite families of modular differential equations.
We show that it is undecidable whether a system of linear equations over the Laurent polynomial ring $\mathbb{Z}[X^{\pm}]$ admit solutions where a specified subset of variables take value in the set of monomials $\{X^z \mid z \in…
We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry…
We introduce the notion of a powerfully solvable group. These are powerful groups possessing an abelian series of a special kind. These groups include in particular the class of powerfully nilpotent groups. We will also see that for a…
We provide an explicit construction for a complete set of orthogonal primitive idempotents of finite group algebras over nilpotent groups. Furthermore, we give a complete set of matrix units in each simple epimorphic image of a finite group…
We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that with the only specific exception the solvable radical of a nonsolvable finite group…