Nilpotent Groups Of Finite Rank Research Articles

Pólya–Carlson dichotomy for coincidence Reidemeister zeta functions via profinite completions

We consider coincidence Reidemeister zeta functions for tame endomorphism pairs of nilpotent groups of finite rank, shedding new light on the subject by means of profinite completion techniques.In particular, we provide a closed formula for coincidence Reidemeister numbers for iterations of endomorphism pairs of torsion-free nilpotent groups of finite rank, based on a weak commutativity condition, which derives from simultaneous triangularisability on abelian sections. Furthermore, we present results in support of a Pólya–Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of the zeta functions in question.

On the lower central series of the free nilpotent groups of finite rank

In their article, “On the derived subgroup of the free nilpotent groups of finite rank” R. D. Blyth, P. Moravec, and R. F. Morse describe the structure of the derived subgroup of a free nilpotent group of finite rank n as a direct product of a nonabelian group and a free abelian group, each with a minimal generating set of cardinality that is a given function of n. They apply this result to computing the nonabelian tensor squares of the free nilpotent groups of finite rank. We generalize their main result to investigate the structure of the other terms of the lower central series of a free nilpotent group of finite rank, each again described as a direct product of a nonabelian group and a free abelian group. In order to compute the ranks of the free abelian components and the size of minimal generating sets for the nonabelian components we introduce what we call weight partitions.

The q-tensor square of finitely generated nilpotent groups, q odd

In the present paper, the authors extend to the [Formula: see text]-tensor square [Formula: see text] of a group [Formula: see text], [Formula: see text] an odd positive integer, some structural results due to Blyth, Fumagalli and Morigi concerning the non-abelian tensor square [Formula: see text] ([Formula: see text]). The results are applied to the computation of [Formula: see text] for finitely generated nilpotent groups [Formula: see text], specially for free nilpotent groups of finite rank. We also generalize to all [Formula: see text] results of Bacon regarding an upper bound to the minimal number of generators of the non-abelian tensor square [Formula: see text] when [Formula: see text] is a [Formula: see text]-generator nilpotent group of class 2. We end by computing the [Formula: see text]-tensor squares of the free [Formula: see text]-generator nilpotent group of class 2, [Formula: see text]. This shows that the above mentioned upper bound is also achieved for these groups when [Formula: see text] odd.

Algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups

We study algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups. A special attention is paid to free nilpotent groups and the groups UTn(Z) of unitriangular (n×n)-matrices over the ring Z of integers for arbitrary n. We observe that the sets of retracts of finitely generated nilpotent groups coincides with the sets of their algebraically closed subgroups. We give an example showing that a verbally closed subgroup in a finitely generated nilpotent group may fail to be a retract (in the case under consideration, equivalently, fail to be an algebraically closed subgroup). Another example shows that the intersection of retracts (algebraically closed subgroups) in a free nilpotent group may fail to be a retract (an algebraically closed subgroup) in this group. We establish necessary conditions fulfilled on retracts of arbitrary finitely generated nilpotent groups. We obtain sufficient conditions for the property of being a retract in a finitely generated nilpotent group. An algorithm is presented determining the property of being a retract for a subgroup in free nilpotent group of finite rank (a solution of a problem of Myasnikov). We also obtain a general result on existentially closed subgroups in finitely generated torsion-free nilpotent with cyclic center, which in particular implies that for each n the group UTn(Z) has no proper existentially closed subgroups.

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ℚ-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.

Existentially Closed Subgroups of Free Nilpotent Groups

Let $$ {N_{\mathrm{c}}} $$ be a variety of all nilpotent groups of class at most c, and let Nr,c be a free nilpotent group of finite rank r and nilpotency class c. It is proved that a subgroup N of Nr,c for c ≥ 3 is existentially closed in Nr,c iff N is a free factor of the group Nr,c with respect to the variety $$ {N_{\mathrm{c}}} $$ . Consequently, N ≃ Nm,c, 1 ≤ m ≤ r, and m ≥ c − 1.

Locally Indicable Groups of Finite Mal’tsev Rank

Recall that G is called a group of finite Mal’tsev rank r if every finitely generated subgroup of G is a group with r generators and r is the least number with this property. Below by the word ‘rank’ we mean the Mal’tsev rank. The notation and terminology adopted here are borrowed from [1, 2]. It is well known that the class of groups of finite rank is closed under subgroups, homomorphic images, and direct products of finitely many groups. For torsion-free Abelian groups, the standard concept of the rank of an Abelian group coincides with that of the Mal’tsev rank. We are also well aware that every torsion-free locally nilpotent group of finite rank is nilpotent and its nilpotency class does not exceed its rank. Groups of finite rank with additional restrictions were studied by many mathematicians such as S. N. Chernikov, V. S. Charin, and D. I. Zaitsev. Research on groups of finite rank was reduced, as a rule, to imposing on a group conditions close to solvability. An impetus for writing this paper is the question whether every linearly orderable finitely generated group will be solvable. The structure of solvable matrix groups over Q is well known. By a theorem of A. I. Mal’tsev, these are finite extensions of triangular matrix groups over Q. We know that every torsion-free solvable group of finite rank can be represented by matrices over the field of rational numbers [1]. Note also that if the rank of a torsion-free solvable matrix group over Q is finite then it is equal to the rational rank (D. I. Zaitsev). In looking into the above question in detail, it became clear that its resolution reduces to treating a similar question for a wider class of groups, namely, the class of locally indicable groups. Recall that a group G is said to be locally indicable if every nontrivial finitely generated subgroup of G admits a homomorphism onto an infinite cyclic group. This class of groups has interesting group-theoretic properties. In [3], it was proved that every one-relator group having no elements of finite order is locally indicable.

Some Residual Properties of Finite Rank Groups

The generalization of one classical Seksenbaev theorem for polycyclic groups is obtained. Seksenbaev proved that if G is a polycyclic group which is residually finite p -group for infinitely many primes p , it is nilpotent. Recall that a group G is said to be a residually finite p -group if for every nonidentity element a of G there exists a homomorphism of the group G onto a finite p -group such that the image of the element a differs from 1. One of the generalizations of the notation of a polycyclic group is the notation of a finite rank group. Recall that a group G is said to be a group of finite rank if there exists a positive integer r such that every finitely generated subgroup in G is generated by at most r elements. We prove the following generalization of Seksenbaev theorem: if G is a group of finite rank which is a residually finite p -group for infinitely many primes p , it is nilpotent. Moreover, we prove that if for every set π of almost all primes the group G of finite rank is a residually finite nilpotent π -group, it is nilpotent. For nilpotent groups of finite rank the necessary and sufficient condition to be a residually finite π -group is obtained, where π is a set of primes.

Verbally and existentially closed subgroups of free nilpotent groups

Let $ {{\mathcal{N}}_c} $ be the variety of all nilpotent groups of class at most c and N r,c a free nilpotent group of finite rank r and nilpotency class c. It is proved that a subgroup H of N r,c (r, c ≥ 1) is verbally closed iff H is a free factor (or, equivalently, an algebraically closed subgroup, a retract) of the group N r,c . In addition, for c ≥ 4 and m < c− 1, every free factor N m,c of the group N c-1,c in the variety $ {{\mathcal{N}}_c} $ is not existentially closed in the group N m+i,c for i = 1, 2,…. It is stated that for r ≥ 3 and 2 ≤ c ≤ 3, every free factor N m,c , 2 ≤ m ≤ r, in $ {{\mathcal{N}}_c} $ is existentially closed in the group N r,c .

Groups elementarily equivalent to a free nilpotent group of finite rank

In this paper, we give a complete algebraic description of groups elementarily equivalent to the P. Hall completion of a given free nilpotent group of finite rank over an arbitrary binomial domain. In particular, we characterize all groups elementarily equivalent to a free nilpotent group of finite rank.

Some structural results on the non-abelian tensor square of groups

We study the non-abelian tensor square Gn G for the class of groups G that are finitely generated modulo their derived subgroup. In particular, we find conditions on G=G 0 so that Gn G is isomorphic to the direct product of 'ðGÞ and the non-abelian exterior square G ^ G. For any group G, we characterize the non-abelian exterior square G ^ G in terms of a presentation of G. Finally, we apply our results to some classes of groups, such as the classes of free solvable and free nilpotent groups of finite rank, and some classes of finite p-groups.

On Primitive Representations of Minimax Nilpotent Groups

We show, in particular, that in the class of minimax two-step nilpotent groups only finitely generated groups can admit exact irreducible primitive representations over a finitely generated field of characteristic zero. We also suggest some approaches to studying induced representations of nilpotent groups of finite rank.

Right-ordered and free lattice-ordered groups

The free l-groups over nilpotent groups are considered. The upper bound of l-solvable length is found for an l-group over a nilpotent group of finite rank; the lower bound of solvable length is given for a free l-group over a finitely generated nilpotent group.

Characteristic subgroups of relatively free groups

A simple new proof is given of a result of Vaughan-Lee which implies that if G is a relatively free nilpotent group of finite rank k and nilpotency class c with c &lt; k then the characteristic subgroups of G are all fully invariant. It is proved that the condition c &lt; k can be weakened to c &lt; k + p − 2 when G has p–power exponent for some prime p. On the other hand it is shown that for each prime p there is a 2-generator relatively free p-group G which is nilpotent of class 2p such that the centre of G is not fully invariant.

Residual finiteness of descending HNN-extension of groups

The paper examines the special case of the general construction of HNN-extensions of groups in which at least one of the associated subgroups is the base group. A criterion is determined for a group obtained in this way to be residually finite. Any group obtained as such an extension from a free nilpotent group of finite rank is residually finite.

Decision problems concerning S-arithmetic groups

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases: = {finitely generated nilpotent groups}; = {(not necessarily associative) rings whose additive group is finitely generated}; = {finitely Z-generated modules over a fixed finitely generated ring}.Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

Some Remarks on Locally Nilpotent Groups of Finite Rank

Some Remarks on Locally Nilpotent Groups of Finite Rank

Some remarks on locally nilpotent groups of finite rank

The main subclasses of locally nilpotent groups are considered in conjunction with the extra hypothesis of finite rank or an allied property. These classes are ordered by inclusion, examples being indicated wherever inclusion is proper.

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].