scholarly journals The Relatively Free Groups F(Nc∧A2) Satisfy Noncentral Commutative Transitivity

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Anthony M. Gaglione ◽  
Seymour Lipschutz ◽  
Dennis Spellman
Keyword(s):  
Free Group ◽  
Model Theory ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Classical Theorem ◽  
Noncentral Element

We prove that a free group, F(Nc∧A2), relative to the variety, Nc∧A2, of all groups simultaneously nilpotent of class at most c and metabelian is such that the centralizer of every noncentral element is abelian. We relate that result to the model theory of such groups as well as a quest to find a relative analog in Nc∧A2 of a classical theorem of Benjamin Baumslag. We also touch briefly on similar considerations in the varieties Nc of nilpotent groups.

∀-free metabelian groups

1997 ◽  
Vol 62 (1) ◽  
pp. 159-174 ◽  
Author(s):  
Olivier Chapuis
Keyword(s):  
Free Group ◽  
Metabelian Group ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Explicit Description ◽  
Universal Theory ◽  
Metabelian Groups ◽  
Free Metabelian Group ◽  
Finitely Generated Groups

In 1965, during the first All-Union Symposium on Group Theory, Kargapolov presented the following two problems: (a) describe the universal theory of free nilpotent groups of class m; (b) describe the universal theory of free groups (see [18, 1.28 and 1.27]). The first of these problems is still open and it is known [25] that a positive solution of this problem for an m ≤ 2 should imply the decidability of the universal theory of the field of the rationals (this last problem is equivalent to Hilbert's tenth problem for the field of the rationals which is a difficult open problem; see [17] and [20] for discussions on this problem). Regarding the second problem, Makanin proved in 1985 that a free group has a decidable universal theory (see [15] for stronger results), however, the problem of deriving an explicit description of the universal theory of free groups is open. To try to solve this problem Remeslennikov gave different characterization of finitely generated groups with the same universal theory as a noncyclic free group (see [21] and [22] and also [11]). Recently, the author proved in [8] that a free metabelian group has a decidable universal theory, but the proof of [8] does not give an explicit description of the universal theory of free metabelian groups.

COMMUTATOR SUBGROUPS OF FREE NILPOTENT GROUPS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1021-1031
Author(s):  
N. GUPTA ◽  
I. B. S. PASSI
Keyword(s):  
Nilpotent Group ◽  
Free Group ◽  
Finite Rank ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Finite Chain ◽  
Free Nilpotent Group ◽  
Infinite Rank ◽  
Derived Subgroup ◽  
Finite Set

For fixed m, n ≥ 2, we examine the structure of the nth lower central subgroup γn(F) of the free group F of rank m with respect to a certain finite chain F = F(0) > F(1) > ⋯ > F(l-1) > F(l) = {1} of free groups in which F(k) is of finite rank m(k) and is contained in the kth derived subgroup δk(F) of F. The derived subgroups δk(F/γn(F)) of the free nilpotent group F/γn(F) are isomorphic to the quotients F(k)/(F(k) ∩ γn(F)) and admit presentations of the form 〈xk,1,…,xk,m(k): γ(n)(F(k))〉, where γ(n)(F(k)), contained in γn(F), is a certain partial lower central subgroup of F(k). We give a complete description of γn(F) as a staggered product Π1 ≤ k ≤ l-1(γ〈n〉(F(k))*γ[n](F(k)))F(k+1), where γ〈n〉(F(k)) is a free factor of the derived subgroup [F(k),F(k)] of F(k) having countable infinite rank and generated by a certain set of reduced commutators of weight at least n, and γ[n](F(k)) is the subgroup generated by a certain finite set of products of non-reduced ordered commutators of weight at least n. There are some far-reaching consequences.

scholarly journals Recognizing powers in nilpotent groups and nilpotent images of free groups

2007 ◽  
Vol 83 (2) ◽  
pp. 149-156
Author(s):  
Gilbert Baumslag
Keyword(s):  
Nilpotent Group ◽  
Free Group ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Factor Group ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Proper Power ◽  
Torsion Free

AbstractAn element in a free group is a proper power if and only if it is a proper power in every nilpotent factor group. Moreover there is an algorithm to decide if an element in a finitely generated torsion-free nilpotent group is a proper power.

scholarly journals The laws of some nilpotent groups of small rank

1974 ◽  
Vol 17 (2) ◽  
pp. 129-132 ◽  
Author(s):  
T. C. Chau
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Small Rank ◽  
Image Position

We shall take for granted the basic terminology currently in use in the theory of varieties of groups. Kovács, Newman, Pentony [2] and Levin [3] prove that if m is an integer greater than 2, then the variety Νm of all nilpotent groups of class at most m is generated by its free group Fm-1(Νm) of rank m – 1 but not by its free group Fm–2(Νm) of rank m — 2. That is, the free groups Fk(Nm), 2≦k ≦ m – 2, do not generate Nm. In general little is known of the varieties generated by them. The purpose of the present paper is to record the varieties of the free groups Fk(Nm) of the nilpotent varieties Nm of all nilpotent groups of class at most m for 2 ≦ k ≦ m – 2 and 5 ≦ m ≦ 6. This is done by describing a basis for the laws in these groups, that is a set of laws the fully invariant closure of which is the set of all laws for Fk(Nm). The set of laws, which, together with the appropriate nilpotency law, form a basis for the relevant groups Fk(Nm) are listed below: .

scholarly journals Generators for the bounded automorphisms of infinite-rank free nilpotent groups

1989 ◽  
Vol 40 (2) ◽  
pp. 175-187 ◽  
Author(s):  
R.G. Burns ◽  
Lian Pi
Keyword(s):  
Nilpotent Group ◽  
Free Group ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Free Nilpotent Group ◽  
Infinite Rank

It is shown that the natural generalisations of the elementary Nielsen transformations of a free group to the infinite-rank case, furnish generators for the subgroup of “bounded” automorphisms of any relatively free nilpotent group of infinite rank. This settles the nilpotent analogue of a question of D. Solitar concerning the “bounded” automorphisms of absolutely free groups of infinite rank.

Subgroups of Finite Index in Free Groups

1949 ◽  
Vol 1 (2) ◽  
pp. 187-190 ◽  
Author(s):  
Marshall Hall
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Recursion Formula ◽  
Independent Interest ◽  
Finiteness Conditions ◽  
Total Length ◽  
Free Generators

This paper has as its chief aim the establishment of two formulae associated with subgroups of finite index in free groups. The first of these (Theorem 3.1) gives an expression for the total length of the free generators of a subgroup U of the free group Fr with r generators. The second (Theorem 5.2) gives a recursion formula for calculating the number of distinct subgroups of index n in Fr.Of some independent interest are two theorems used which do not involve any finiteness conditions. These are concerned with ways of determining a subgroup U of F.

scholarly journals A dendrological proof of the Scott conjecture for automorphisms of free groups

1998 ◽  
Vol 41 (2) ◽  
pp. 325-332 ◽  
Author(s):  
D. Gaboriau ◽  
G. Levitt ◽  
M. Lustig
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Alternative Proof ◽  
Automorphisms Of Free Groups

Let α be an automorphism of a free group of rank n. The Scott conjecture, proved by Bestvina-Handel, asserts that the fixed subgroup of α has rank at most n. We give a short alternative proof of this result using R-trees.

scholarly journals On a Nonsingular Equation of Length 9 Over Torsion Free Groups

2019 ◽  
Vol 12 (2) ◽  
pp. 590-604
Author(s):  
M. Fazeel Anwar ◽  
Mairaj Bibi ◽  
Muhammad Saeed Akram
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Torsion Free

In \cite{levin}, Levin conjectured that every equation is solvable over a torsion free group. In this paper we consider a nonsingular equation $g_{1}tg_{2}t g_{3}t g_{4} t g_{5} t g_{6} t^{-1} g_{7} t g_{8}t \\ g_{9}t^{-1} = 1$ of length $9$ and show that it is solvable over torsion free groups modulo some exceptional cases.

scholarly journals Centralisers of Dehn twist automorphisms of free groups

2015 ◽  
Vol 159 (1) ◽  
pp. 89-114 ◽  
Author(s):  
MORITZ RODENHAUSEN ◽  
RICHARD D. WADE
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Dehn Twist ◽  
Classifying Space ◽  
Dehn Twists ◽  
Automorphisms Of Free Groups

AbstractWe refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an explicit presentation for the centraliser of a Nielsen automorphism in Aut(Fn). This gives restrictions to actions of Aut(Fn) on CAT(0) spaces.

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