Solving equations in free nilpotent groups
نویسندگان
چکیده
In this paper we show that any system of equations over a free nilpotent group of class c is either unitary or nullary. In fact, such a system either has a most general solution (akin to the most general solution of a system of linear diophantine equations), or every solution has a proper generalization. In principle we provide an algorithm for determining whether or not a most general solution exists, and exhibiting it if it does. This requires solving a system of linear diophantine equations for approximately c(^j different right hand sides, where k is the number of parameters which occur in a most general solution of the system in the variety of abelian groups.
منابع مشابه
Local Functional Equations for Submodule Zeta Functions Associated to Nilpotent Algebras of Endomorphisms
We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on such functional equations for ideal zeta functions of nilpotent Lie lattices. Via the Mal’cev correspondence, these results have corollaries pertaining to ze...
متن کاملFunctional equations for local normal zeta functions of nilpotent groups
We give an explicit formula for the local normal zeta functions of torsion-free, class-2-nilpotent groups at primes p where the associated Pfaffian hypersurface has good reduction mod p and contains no lines. We show how together the functional equations of two types of zeta functions the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by ...
متن کامل(c,1,...,1) Polynilpotent Multiplier of some Nilpotent Products of Groups
In this paper we determine the structure of (c,1,...,1) polynilpotent multiplier of certain class of groups. The method is based on the characterizing an explicit structure for the Baer invariant of a free nilpotent group with respect to the variety of polynilpotent groups of class row (c,1,...,1).
متن کاملNormal subgroup growth in free class-2-nilpotent groups
Let F2,d denote the free class-2-nilpotent group on d generators. We compute the normal zeta functions ζ F2,d(s), prove that they satisfy local functional equations and determine their abscissae of convergence and pole orders.
متن کاملResults on Engel Fuzzy Subgroups
In the classical group theory there is an open question: Is every torsion free n-Engel group (for n ≥ 4), nilpotent?. To answer the question, Traustason [11] showed that with some additional conditions all 4-Engel groups are locally nilpotent. Here, we gave some partial answer to this question on Engel fuzzy subgroups. We show that if μ is a normal 4-Engel fuzzy subgroup of ...
متن کامل