Extended genus of torsion-free finitely generated nilpotent groups of class two

The extended genus of a nilpotent group N is the set of isomorphism classes of nilpotent groups M, not necessarily finitely generated, such that the p-localizations M _p , N _p are isomorphic for all primes p. In this article, for any torsion-free finitely generated nilpotent group N of nilpotency class 2, the extended genus of N is analyzed by assigning to each of its members a sequence of triads of matrices with rational entries, generalizing the sequential representation which has been exploited elsewhere in the case when N is abelian. This approach allows, among other things, to obtain examples of groups in the ordinary (Mislin) genus of N     

On stable torsion-free nilpotent groups

Summary We show that an infinite field is interpretable in a stable torsion-free nilpotent groupG of classk, k>1. Furthermore we prove thatG/Z _k-1 (G) must be divisible. By generalising methods of Belegradek we classify some stable torsion-free nilpotent groups modulo isomorphism and elementary equivalence.Supported by the Deutsche Forschungsgemeinschaft     

On multipliers of torsion-free nilpotent groups

The notion of the Schur multiplier is carried over to torsion-free nilpotent groups of finite rank, and the relation between the rank of a torsion-free nilpotent group and the rank of its multiplier is determined, [3].Translated from Matematicheskie Zametki, Vol. 5, No. 5, pp. 541–544, May, 1969.     

On quasivarieties of axiomatic rank 3 of torsion-free nilpotent groups

We study the lattice of quasivarieties of axiomatic rank at most 3 of torsion-free nilpotent groups of class at most 3. We prove that this lattice has cardinality of the continuum and includes a sublattice that is order isomorphic to the set of real numbers. Also we establish that the lattice of quasivarieties of axiomatic rank at most 2 of these groups is a 5-element chain.     

The arithmetical hierarchy of nilpotent torsion-free groups

Previously, N. Khisamiev proved that all {ie172-1} Abelian torsion-free groups are {ie172-2}. We prove that for the class of nilpotent torsion-free groups, the situation is different: even the quotient group F of a {ie172-3} nilpotent group of class 2 by its periodic part may fail to have a {ie172-4}. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 308–313, May–June, 1996.     

Computable torsion-free nilpotent groups of finite dimension

We find criteria for the computability (constructivizability) of torsion-free nilpotent groups of finite dimension. We prove the existence of a principal computable enumeration of the class of all computable torsion-free nilpotent groups of finite dimension. An example is constructed of a subgroup in the group of all unitriangular matrices of dimension 3 over the field of rationals that is not computable but the sections of any of its central series are computable.     

Irreducible representations of finitely generated nilpotent torsion-free groups

An investigation of the structure of the quotient algebra, with respect to a prime ideal, of the group algebra of a finitely-generated nilpotent torsion-free group. Conditions are studied under which an irreducible representation of such a group is induced.Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 199–210, February, 1971.     

Classification of torsion-free genus zero congruence groups

We study and classify all torsion-free genus zero congruence subgroups of the modular group.

     

On odd order nilpotent groups with class 2

Let G be an odd order nilpotent group with class 2 and let e denote the exponent of its commutator subgroup. Let ${e=p_1^{r_1}p_2^{r_2}\ldots p_s^{r_s}}Let G be an odd order nilpotent group with class 2 and let e denote the exponent of its commutator subgroup. Let e=p₁^r₁p₂^r₂?p_s^r_s{e=p_1^{r_1}p_2^{r_2}\ldots p_s^{r_s}}, where the p _i ’s are the prime divisors of the order of G and the r _i ’s are non-negative integers. Then there are at least (?_i=1^s(1+r_i))-1{\left(\prod_{i=1}^{s}(1+r_i)\right)-1} non-isomorphic nilpotent groups with class 2 and each of the groups has the same order structure as G.