scholarly journals On palindromic width of certain extensions and quotients of free nilpotent groups

2014 ◽  
Vol 24 (05) ◽  
pp. 553-567 ◽  
Author(s):  
Valeriy G. Bardakov ◽  
Krishnendu Gongopadhyay
Keyword(s):  
Lower Bound ◽  
Normal Subgroup ◽  
Nilpotent Group ◽  
Metabelian Group ◽  
Nilpotent Groups ◽  
Free Metabelian Group ◽  
Free Nilpotent Group

In [Bardakov and Gongopadhyay, Palindromic width of free nilpotent groups, J. Algebra 402 (2014) 379–391] the authors provided a bound for the palindromic widths of free abelian-by-nilpotent group ANn of rank n and free nilpotent group N n,r of rank n and step r. In the present paper, we study palindromic widths of groups [Formula: see text] and [Formula: see text]. We denote by [Formula: see text] the quotient of the group Gn = 〈x1, …, xn〉, which is free in some variety by the normal subgroup generated by [Formula: see text]. We prove that the palindromic width of the quotient [Formula: see text] is finite and bounded by 3n. We also prove that the palindromic width of the quotient [Formula: see text] is precisely 2(n - 1). As a corollary to this result, we improve the lower bound of the palindromic width of N n,r. We also improve the bound of the palindromic width of a free metabelian group. We prove that the palindromic width of a free metabelian group of rank n is at most 4n - 1.

scholarly journals DISTORTION IN FREE NILPOTENT GROUPS

2010 ◽  
Vol 20 (05) ◽  
pp. 661-669 ◽  
Author(s):  
TARA C. DAVIS
Keyword(s):  
Nilpotent Group ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Nilpotent Group

We prove that a subgroup of a finitely generated free nilpotent group F is undistorted if and only if it is a retract of a subgroup of finite index in F.

The Structure of 𝐺-Free Nilpotent Groups of Step 3

2004 ◽  
Vol 11 (1) ◽  
pp. 27-33
Author(s):  
M. Amaglobeli
Keyword(s):  
Nilpotent Group ◽  
Canonical Form ◽  
Nilpotent Groups ◽  
Free Nilpotent Group

Abstract The canonical form of elements of a 𝐺-free nilpotent group of step 3 is defined assuming that the group 𝐺 contains no elements of order 2.

The Group Ring Of a Class Of Infinite Nilpotent Groups

1955 ◽  
Vol 7 ◽  
pp. 169-187 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Nilpotent Group ◽  
Group Ring ◽  
Characteristic Zero ◽  
Discrete Group ◽  
Nilpotent Groups ◽  
Zero Element ◽  
Free Nilpotent Group ◽  
Image Position ◽  
Torsion Free ◽  
The Ideal

Introduction. In this paper we study the (discrete) group ring Γ of a finitely generated torsion free nilpotent group over a field of characteristic zero. We show that if Δ is the ideal of Γ spanned by all elements of the form G − 1, where G ∈ , thenand the only element belonging to Δw for all w is the zero element (cf. (4.3) below).

scholarly journals Fitting quotients of finitely presented abelian-by-nilpotent groups

2014 ◽  
Vol 17 (1) ◽  
pp. 1-12
Author(s):  
J. R. J. Groves ◽  
Ralph Strebel
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Finitely Presented

Abstract.We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

∀-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 Commutators and Squares in Free Nilpotent Groups

2009 ◽  
Vol 2009 ◽  
pp. 1-10
Author(s):  
Mehri Akhavan-Malayeri
Keyword(s):  
Nilpotent Group ◽  
Free Group ◽  
Nilpotent Groups ◽  
Free Nilpotent Group ◽  
Rank 2

In a free group no nontrivial commutator is a square. And in the free groupF2=F(x1,x2)freely generated byx1,x2the commutator[x1,x2]is never the product of two squares inF2, although it is always the product of three squares. LetF2,3=〈x1,x2〉be a free nilpotent group of rank 2 and class 3 freely generated byx1,x2. We prove that inF2,3=〈x1,x2〉, it is possible to write certain commutators as a square. We denote bySq(γ)the minimal number of squares which is required to writeγas a product of squares in groupG. And we defineSq(G)=sup{Sq(γ);γ∈G′}. We discuss the question of when the square length of a given commutator ofF2,3is equal to 1 or 2 or 3. The precise formulas for expressing any commutator ofF2,3as the minimal number of squares are given. Finally as an application of these results we prove thatSq(F′2,3)=3.

Generalized partition functions and subgroup growth of free products of nilpotent groups

Analysis ◽  
2005 ◽  
Vol 25 (4) ◽  
Author(s):  
Thomas W. Müller ◽  
Jan-Christoph Schlage-Puchta
Keyword(s):  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Partition Functions ◽  
Free Products ◽  
Subgroup Growth ◽  
Free Nilpotent Group ◽  
Generalized Partition ◽  
Torsion Free

SummaryWe estimate the growth of homomorphism numbers of a torsion-free nilpotent group

Automorphism sequences of free nilpotent groups of class two

1976 ◽  
Vol 79 (2) ◽  
pp. 271-279 ◽  
Author(s):  
Joan L. Dyer ◽  
Edward Formanek
Keyword(s):  
Automorphism Group ◽  
Nilpotent Group ◽  
Finite Rank ◽  
Nilpotent Groups ◽  
Split Extension ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Image Position

In this paper we prove that the automorphism group A(N) of a free nilpotent group N of class 2 and finite rank n is complete, except when n is 1 or 3. Equivalently, the centre of A(N) is trivial and every automorphism of A(N) is inner, provided n ≠ 1 or 3. When n = 3, A(N) has an our automorphism of order 2, so A(A(N)) is a split extension of A(N) by . In this case, A(A(N)) is complete. These results provide some evidence supporting a conjecture of Gilbert Baumslag that the sequencebecomes periodic if N is a finitely generated nilpotent group.

SUBSPACES, FREE SUBGROUPS AND A THEOREM OF STALLINGS

2006 ◽  
Vol 16 (03) ◽  
pp. 475-491
Author(s):  
GILBERT BAUMSLAG
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Nilpotent Group ◽  
Homology Group ◽  
Integral Homology ◽  
Free Nilpotent Group ◽  
Theoretic Proof ◽  
Analogous Formula ◽  
Free Subgroups ◽  
Torsion Free

There is a simple group-theoretic formula for the second integral homology group of a group. This is an abelian group and there is an analogous formula for another abelian group, which involves a normal subgroup N of a torsion-free nilpotent group G. Properties of this abelian group translate into properties of G/N. This approach allows one to give a simple purely group-theoretic proof of an old theorem of J. R. Stallings, namely that if Γ is a group, if H1(G,ℤ) is free abelian and if H2(G,ℤ) = 0, then any subset Y of G which is independent modulo the derived group of G, freely generates a free group. The ideas used admit to considerable generalization, yielding in particular, proofs of a number of theorems of U. Stammbach.

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