Torsion-free Nilpotent Groups Of Class Research Articles

On Dominions of the Rationals in Nilpotent Groups

The dominion of a subgroup H of a group G in a class M is the set of all a ∈ G that have the same images under every pair of homomorphisms, coinciding on H from G to a group in M. A group H is n-closed in M if for every group G = gr(H, a1,..., an) in M that includes H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rationals is 2-closed in every quasivariety of torsion-free nilpotent groups of class at most 3.

The power graph of a torsion-free group

The \emph{power graph} $P(G)$ of a group $G$ is the graph whose vertex set is $G$, with $x$ and $y$ joined if one is a power of the other; the \emph{directed power graph} $\vec{P}(G)$ has the same vertex set, with an arc from $x$ to $y$ if $y$ is a power of $x$. It is known that, for finite groups, the power graph determines the directed power graph up to isomorphism. However, it is not true that any isomorphism between power graphs induces an isomorphism between directed power graphs. Moreover, for infinite groups the power graph may fail to determine the directed power graph. In this paper, we consider power graphs of torsion-free groups. Our main results are that, for torsion-free nilpotent groups of class at most $2$, and for groups in which every non-identity element lies in a unique maximal cyclic subgroup, the power graph determines the directed power graph up to isomorphism. For specific groups such as $\mathbb{Z}$ and $\mathbb{Q}$, we obtain more precise results. Any isomorphism $P(\mathbb{Z})\to P(G)$ preserves orientation, so induces an isomorphism between directed power graphs; in the case of $\mathbb{Q}$, the orientations are either all preserved or all reversed. We also obtain results about groups in which every element is contained in a unique maximal cyclic subgroup (this class includes the free and free abelian groups), and about subgroups of the additive group of $\mathbb{Q}$ and about $\mathbb{Q}^n$.

On 2-Closedness of the Rational Numbers in Quasivarieties of Nilpotent Groups

The dominion of a subgroup H of a group G in a class M is the set of all elements a ∈ G that have equal images under every pair of homomorphisms from G to a group of M coinciding on H. A group H is said to be n-closed in M if for every group G = gr(H, a1,..., a n ) of M that contains H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rational numbers is 2-closed in every quasivariety M of torsion-free nilpotent groups of class at most 3 whenever every 2-generated group of M is relatively free.

Small doubling in ordered nilpotent groups of class 2

The aim of this paper is to present a complete description of the structure of finite subsets S of torsion-free nilpotent groups of class 2 satisfying |S2|=3|S|−2. In view of results in [12], this gives a complete description of the structure of finite subsets with the above property in any torsion-free nilpotent group.

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.

Dominions of universal algebras and projective properties

Let A be a universal algebra and H its subalgebra. The dominion of H in A (in a class {ie304-01}) is the set of all elements a ? A such that every pair of homomorphisms f, g: A ? ? {ie304-02} satisfies the following: if f and g coincide on H, then f(a) = g(a). A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras H whose dominions coincide with H. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup H is closed in each group ?H, a? generated by one element modulo H.

Decompositions of finitely generated nilpotent groups of class 2

In this paper we prove that rational indecomposability is a genus property for finitely generated torsion-free nilpotent groups of class 2. We use this result to determine the genus of finitely generated torsion-free nilpotent groups of class 2 which decompose as a direct product of rationally indecomposable groups.

Characters of the discrete Heisenberg group and of its completion

In this paper we describe the relationship between characters of finitely generated torsion-free nilpotent groups of class 2 and their completions. In particular we give a detailed description of this connection for the discrete Heisenberg group.

Isoperimetric functions of nilpotent groups amalgamated along a cyclic subgroup

We show that amalgams of finitely generated torsionfree nilpotent groups of class≤c along a cyclic subgroup satisfy a polynomial isoperimetric inequality of degree at most 4c2. The distortion of the amalgamated subgroup is bounded above by a polynomial of degree c.We also give an example of a non‐cyclic amalgam of finitely generated torsionfree nilpotent groups along an abelian, isolated and normal subgroup having an exponential isoperimetric inequality.

On genus and embeddings of torsion-free nilpotent groups of class two

We study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings between members of each Mislin genus. In this sense, the family Γ resembles the family N0 of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with N0, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes. We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent groups of class 2 nor for torsion-free abelian groups of finite rank.

Polynomiality Properties of Group Extensions with a Torsion-Free Abelian Kernel

In studying torsion-free nilpotent groups of class 2 it is a key fact that each central group extension Z n ↣ E↠ Z m is representable by a bilinearcocycle. So for investigating nilpotent groups of higher class it is natural to ask for a generalization of this fact, namely when Z m is replaced by some torsion-free nilpotentgroup Gand Z n by some torsion-free abeliangroup Bwith nilpotent G-action. In this paper we study a suitable notion of ( bi) polynomialcocycles (in the strong sense of polynomiality introduced by Passi) and prove the desired representability theorem(4.3). This was known before only for centralextensions with divisiblekernel and with kernel Z if Gis abelian, nilpotent of class 2 or if the quotients of the lower central series of Gare torsion-free. Our representability result implies a convergence theoremfor an approximation of H 2( G, B) by polynomial cohomology groups(4.4). Since the latter ones are well accessible to computation one obtains a formula for H 2( G, B) in terms of a presentation of Gwhich can be evaluated by integral matrix calculus. More precisely, a given presentation of Gamounts to a three-term cochain complex consisting of finitely generated free Z -modules if the groups Gand Bare finitely generated; the cohomology of this complex is identified with H 2( G, B) in such a way that representing 2-cocycles are explicitly given in terms of integer valued rational polynomial functions (6.4). As an application, we establish an explicit bijectionbetween torsion-free nilpotent groupsand Z -torsion-free nilpotent Lie ringsboth finitely generated and nilpotent of class≤3 (7.2). In case a given group extension is representable by a polynomial cocycle we also determine the minimal degreeof polynomiality for which this holds. Indeed, dropping the assumption that Gis torsion-free nilpotent, we give an intrinsic characterizationof all group extensions with torsion-free abelian kernel which are representable by a polynomial cocycle of degree ≤ n, provided that the cokernel is finitely generated and acts nilpotently on the kernel (4.2). Thus a polynomiality theory for group extensions as asked for by Passi is now achieved in case the kernel is torsion-free. A motivation for this—apart from calculating H 2( G, B) explicitly—comes from a question posed by J. Milnor in 1977, namely whether all finitely generated, torsion-free virtually polycyclic groups arise as fundamental groups of compact, complete affinely flat manifolds. Even the case of torsion-free nilpotent groups is interesting but far from being understood, notably since counterexamples of this type have recently been discovered. A connection of Milnor's question for nilpotent groups with polynomial constructions was first indicated in the work of P. Igodt and K. B. Lee. Recently a very close connection of this type was established, and an obstruction theorydeveloped for the problem in terms of polynomial cohomology. On the other hand, polynomiality properties of extensions with a torsionkernel are closely related to dimension subgroups, as was observed by Passi in the case of centralextensions. An extension of this idea to the noncentralcase is provided by Theorem 3.4 below. This might be of interest in connection with the recent discovery of Gupta and Kuz'min that the quotient group D n ( G)/γ n ( G) is an abelian normal but in general noncentral subgroup of G/γ n ( G). Moreover, we obtain a functorial equivalencebetween extensions of torsion-free nilpotent groupsand of Z - torsion-free nilpotent modules(5.1). This improves a general result of Reiner and Roggenkamp in the special situation of torsion-free nilpotent groups. As applications, it yields an inductive cohomological description of automorphism groupsof torsion-free nilpotent groups [9] and a generalization of the classical Dold-Kan equivalence—between simplicial abelian groups and chain complexes of abelian groups—to simplicial groups of class 2. Applications in localization theory are also to be expected, namely to questions concerning the genus of torsion-free nilpotent groups. All the above-mentioned results are based on the more technical work of the first section. There, also, generalizations of theorems of Witt and Quillen are obtained concerning certain Lie algebras associated with groups; these results might be of independent interest.

Automorphisms of Torsion-Free Nilpotent Groups of Class Two

We construct $2$-divisible, torsion-free abelian groups $G$ admitting an alternating bilinear map. We use these groups $G$ to find nilpotent groups $N$ of class $2$ such that $\operatorname {Aut}(N)$ modulo a natural normal subgroup is a prescribed group.

Automorphisms of torsion-free nilpotent groups of class two

We construct 2 2 -divisible, torsion-free abelian groups G G admitting an alternating bilinear map. We use these groups G G to find nilpotent groups N N of class 2 2 such that Aut ⁡ ( N ) \operatorname {Aut}(N) modulo a natural normal subgroup is a prescribed group.

On countable locally described structures

We consider classes of locally X-structures of the form LX = {Gƒ∀a 1, … a n ϵ G ∃A ϵ Ч such that α 1, …, a n ϵ A ⩽ G} where Ч is a countable set of isomorphism types of finitely generated structures. In LЧ we study (1) closed structures also known as basically saturated structures which are algebraic analogues of existentially closed structures and (2) saturated structures which are algebraic analogues of existentially universal structures. We do not suppose that LЧ have the amalgamation property as in Jónsson type model theory nor that LЧ be first-order axiomatizable as in classical model theory. The main theorems of the general theory give characterizations for (1) uniqueness, up to isomorphism, of the countable closed structure and (2) the existence of a countable saturated structure. Our criterions are given in terms of local amalgamation properties of Ч. New among the examples are the results that there exists a unique countable closed structure in the classes of locally finitely presented groups, torsion-free locally nilpotent groups, and torsion-free nilpotent groups of class c ( c≥ 2).

Existentially closed torsion-free nilpotent groups of class three

We denote by and the classes of torsion-free nilpotent groups of nilpotency class at most two and three, respectively. In this paper we show that most of the known results about existentially closed (e.c.) groups in remain true in : Up to isomorphism, there exist only countably many countable e.c. groups and they are distinguished by the ranks of their centers. An e.c. group is finitely (infinitely) generic if and only if the center has dimension one (≥ω). Apart from trivial exceptions, e.c, algebraically closed, and “closed with respect to systems of equations in one unknown” are equivalent.Let K be a class of groups. A group G ϵ K is called existentially closed in K if G contains a solution of any finite system Σ of equations and inequations with constants in G and one or more unknowns, provided that Σ has a solution in some in K. If Σ may contain equations (in at most n unknowns) only, G is called algebraically closed (n-unknown closed [1]). We use the abbreviations e.c, a.c and n-u.c. throughout. For more background on these terms the reader is referred to [3] and [4]. We also assume a basic knowledge of generic structures. The main group theoretic notions needed here are explained in §2; [2] and [11] are the general references for this field.

A note on finitely generated torsion-free nilpotent groups of class 2

A note on finitely generated torsion-free nilpotent groups of class 2

Algebraically closed torsion-free nilpotent groups of class 2

Algebraically closed torsion-free nilpotent groups of class 2

Automorphism Groups of Torsion-Free Nilpotent Groups of Class 2

Automorphism Groups of Torsion-Free Nilpotent Groups of Class 2