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.

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.

The isomorphism problem for torsion free nilpotent groups of Hirsch length at most 5

2017 ◽  
Vol 9 (1) ◽  
Author(s):  
Bettina Eick ◽  
Ann-Kristin Engel
Keyword(s):  
Nilpotent Group ◽  
Canonical Form ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Explicit Description ◽  
Isomorphism Type ◽  
Polynomial Equations ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Torsion Free

AbstractWe consider the isomorphism problem for the finitely generated torsion free nilpotent groups of Hirsch length at most five. We show how this problem translates to solving an explicitly given set of polynomial equations. Based on this, we introduce a canonical form for each isomorphism type of finitely generated torsion free nilpotent group of Hirsch length at most 5 and, using a variation of our methods, we give an explicit description of its automorphisms.

scholarly journals Symbolic Collection using Deep Thought

1998 ◽  
Vol 1 ◽  
pp. 9-24 ◽  
Author(s):  
C. R. Leedham-Green ◽  
Leonard H. Soicher
Keyword(s):  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Deep Thought ◽  
Torsion Free

AbstractWe describe the “Deep Thought” algorithm, which can, among other things, take a commutator presentation for a finitely generated torsion-free nilpotent group G, and produce explicit polynomials for the multiplication of elements of G. These polynomials were first shown to exist by Philip Hall, and allow for “symbolic collection” in finitely generated nilpotent groups. We discuss various practicalissues in calculations in such groups, including the construction of a hybrid collector, making use of both the polynomials and ordinary collection from the left.

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.

scholarly journals Decomposability of finitely generated torsion-free nilpotent groups

2016 ◽  
Vol 26 (08) ◽  
pp. 1529-1546 ◽  
Author(s):  
Gilbert Baumslag ◽  
Charles F. Miller ◽  
Gretchen Ostheimer
Keyword(s):  
Nilpotent Group ◽  
Direct Product ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Torsion Free

We describe an algorithm for deciding whether or not a given finitely generated torsion-free nilpotent group is decomposable as the direct product of nontrivial subgroups.

Genus of nilpotent groups of Hirsch length six

1995 ◽  
Vol 117 (3) ◽  
pp. 431-438 ◽  
Author(s):  
Charles Cassidy ◽  
Caroline Lajoie
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Torsion Free

AbstractIn this paper, we characterize the genus of an arbitrary torsion-free finitely generated nilpotent group of class two and of Hirsch length six by means of a finite number of arithmetical invariants. An algorithm which permits the enumeration of all possible genera that can occur under the conditions above is also given.

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 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 Finitely generated nilpotent group C*-algebras have finite nuclear dimension

2018 ◽  
Vol 2018 (738) ◽  
pp. 281-298 ◽  
Author(s):  
Caleb Eckhardt ◽  
Paul McKenney
Keyword(s):  
Irreducible Representation ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Irreducible Representations ◽  
Finitely Generated ◽  
C Algebra ◽  
C Algebras ◽  
K Theory ◽  
Universal Coefficient Theorem ◽  
Torsion Free

Abstract We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra A generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, A satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its ordered K-theory and is approximately subhomogeneous. We observe that all C*-algebras generated by faithful irreducible representations of finitely generated, torsion free nilpotent groups satisfy the universal coefficient theorem.

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