Commutator length of abelian-by-nilpotent groups

1998 ◽  
Vol 40 (1) ◽  
pp. 117-121 ◽  
Author(s):  
Mehri Akhavan-Malayeri ◽  
Akbar Rhemtulla
Keyword(s):  
Nilpotent Group ◽  
Commutator Subgroup ◽  
Nilpotent Groups ◽  
Free Nilpotent Group ◽  
Commutator Length

AbstractLet G be a group and C = [G, G] be its commutator subgroup. Denote by c(G) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. The exact values of c{G) are computed when G is a free nilpotent group or a free abelian-by-nilpotent group. If G is a free nilpotent group of rank n>2 and class c>2, c(G) = [n/2] if c = 2 and c(G) = n if c>2. If G is a free abelian-by-nilpotent group of rank n > 2 then c(G) = n.

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.

scholarly journals On almost finitely generated nilpotent groups

1996 ◽  
Vol 19 (3) ◽  
pp. 539-544 ◽  
Author(s):  
Peter Hilton ◽  
Robert Militello
Keyword(s):  
Nilpotent Group ◽  
Commutator Subgroup ◽  
Local Group ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Finitely Generated

A nilpotent groupGis fgp ifGp, is finitely generated (fg) as ap-local group for all primesp; it is fg-like if there exists a nilpotent fg groupHsuch thatGp≃Hpfor all primesp. The fgp nilpotent groups form a (generalized) Serre class; the fg-like nilpotent groups do not. However, for abelian groups, a subgroup of an fg-like group is fg-like, and an extension of an fg-like group by an fg-like group is fg-like. These properties persist for nilpotent groups with finite commutator subgroup, but fail in general.

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).

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

scholarly journals A test for commutators

1976 ◽  
Vol 17 (1) ◽  
pp. 31-36 ◽  
Author(s):  
Hans Liebeck
Keyword(s):  
Nilpotent Group ◽  
Commutator Subgroup ◽  
Free Nilpotent Group ◽  
Image Position

In the course of a study of commutator subgroups I. D. Macdonald [1] presented the free nilpotent group G4 of class 2 on 4 generators as an example of a nilpotent group whose commutator subgroup has elements that are not commutators. To demonstrate this he proceeded as follows: let G4 = 〈a1, a2, a3, a4〉 and put cij = [ai, aj] for 1 ≦ i < j ≦ 4. Then the relations in G4 are [cij, ak] = 1 for 1 ≦ i < j ≦ 4 and 1 ≦ k ≦ 4, and their consequences. Macdonald observed that an arbitrary commutator may be written aswhich simplifies towhere δij = αiβj - αjβi The indices δij satisfy the relationIt follows that the element c13c24 in G′4 (for which δ12 = δ14 = δ23 = δ34 = 0 and δ13 = δ24 = 1) is not a commutator.

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.

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.

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