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.

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 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 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.

scholarly journals On nilpotent and polycyclic groups

1989 ◽  
Vol 40 (1) ◽  
pp. 119-122
Author(s):  
Robert J. Hursey
Keyword(s):  
Nilpotent Group ◽  
Central Series ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Polycyclic Groups ◽  
Torsion Free

A group G is torsion-free, finitely generated, and nilpotent if and only if G is a supersolvable R-group. An ordered polycylic group G is nilpotent if and only if there exists an order on G with respect to which the number of convex subgroups is one more than the length of G. If the factors of the upper central series of a torsion-free nilpotent group G are locally cyclic, then consecutive terms of the series are jumps, and the terms are absolutely convex subgroups.

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

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