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On computing the nonabelian tensor square of polycyclic groups
"... The nonabelian tensor square G⊗G of the group G is the group generated by the symbols g ⊗ h, where g, h ∈ G, subject to the relations gg ′ ⊗ h = (gg ′ ⊗ gh)(g ⊗ h) and g ⊗ hh ′ = (g ⊗ h)(hg ⊗ hh′) for all g, g, h, h ′ ∈ G, where gg ′ = gg′g−1 is conjugation on the left. Following the work of C. ..."
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The nonabelian tensor square G⊗G of the group G is the group generated by the symbols g ⊗ h, where g, h ∈ G, subject to the relations gg ′ ⊗ h = (gg ′ ⊗ gh)(g ⊗ h) and g ⊗ hh ′ = (g ⊗ h)(hg ⊗ hh′) for all g, g, h, h ′ ∈ G, where gg ′ = gg′g−1 is conjugation on the left. Following the work of C. Miller [18], R. K. Dennis in [10] introduced the nonabelian tensor square which is a specialization of the more general nonabelian tensor product independently introduced by R. Brown and J.L. Loday [9]. By computing the nonabelian tensor square we mean finding a standard or simplified presentation for it. In the case of finite groups, the definition of the nonabelian tensor square gives a finite presentation that can be simplified using Tietze transformations. This simplified presentation can then be examined to determine the nonabelian tensor square. This was the approach taken in [8], in which the nonabelian tensor square was computed for each nonabelian group of order up to 30. Creating a presentation from the definition of the nonabelian tensor square, simplifying it using Tietze transformations and computing a structure description from the simplified presentation can be implemented in few lines of GAP [16]. However, this strategy does not scale well to finite groups G having order greater than 100 since the initial presentation has G2 generators and 2G3 relations. The most general method for computing the nonabelian tensor square uses the notion of a crossed pairing (see [8]). Let G and L be groups. We call the mapping Φ: G×G → L a crossed pairing if for all g, g′, g′ ′ ∈ G we have
On the orbitstabilizer problem for integral matrix actions of polycyclic groups
 Math. Comp. (Number
, 2002
"... Abstract. We present an algorithm to solve the orbitstabilizer problem for a polycyclic group G acting as a subgroup of GL(d, Z) on the elements of Q d. We report on an implementation of our method and use this to observe that the algorithm is practical. 1. ..."
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Abstract. We present an algorithm to solve the orbitstabilizer problem for a polycyclic group G acting as a subgroup of GL(d, Z) on the elements of Q d. We report on an implementation of our method and use this to observe that the algorithm is practical. 1.
A Practical Algorithm for Finding Matrix Representations for Polycyclic Groups
 J. SYMBOLIC COMPUT
, 1997
"... We describe a new algorithm for finding matrix representations for polycyclic groups given by finite presentations. In contrast to previous algorithms, our algorithm is efficient enough to construct representations for some interesting examples. The examples which we studied included a collection of ..."
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We describe a new algorithm for finding matrix representations for polycyclic groups given by finite presentations. In contrast to previous algorithms, our algorithm is efficient enough to construct representations for some interesting examples. The examples which we studied included a collection of free nilpotent groups, and our results here led us to a theoretical result concerning such groups.
Algorithms for Polycyclicbyfinite Groups
, 1996
"... Let R be a number field. We present several algorithms for working with polycyclicbyfinite subgroups of GL(n; R). Let G be a subgroup of GL(n; R) given by a finite generating set of matrices. We describe an algorithm for deciding whether or not G is polycyclicbyfinite. For polycyclicbyfinite ..."
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Let R be a number field. We present several algorithms for working with polycyclicbyfinite subgroups of GL(n; R). Let G be a subgroup of GL(n; R) given by a finite generating set of matrices. We describe an algorithm for deciding whether or not G is polycyclicbyfinite. For polycyclicbyfinite G, we describe an algorithm for deciding whether or not a given matrix is an element of G. We prove that an abstract group G has a faithful representation as a triangularizable subgroup of GL(n; Z) for some n if and only if G is polycyclic and the commutator subgroup of G is torsionfree nilpotent. Suppose G is a polycyclic group given by a consistent polycyclic presentation. We describe an algorithm for deciding whether or not G has a faithful representation as a triangularizable subgroup of GL(n; Z), as well as an algorithm for constructing such a representa...
Rewriting Techniques in Finitely Presented Groups and Monoids
, 1997
"... This document contains an amplified version of the five talks given by the author at the 22nd Holiday Mathematics Symposium at the New Mexico State University, January 37 1997, on the topic "Rewriting Techniques and Noncommutative Gröbner Bases". ..."
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This document contains an amplified version of the five talks given by the author at the 22nd Holiday Mathematics Symposium at the New Mexico State University, January 37 1997, on the topic "Rewriting Techniques and Noncommutative Gröbner Bases".
Investigating selfsimilar groups using their finite Lpresentation
, 2012
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Representing Subgroups of Finitely Presented Groups by Quotient Subgroups
"... This article proposes to represent subgroups of finitely presented 1. Creation of Quotient Representations and Easy Calculations groups by their image in a quotient. It gives algorithms for basic 2. Subdirect Products operations in this representation and investigates how iteration ..."
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This article proposes to represent subgroups of finitely presented 1. Creation of Quotient Representations and Easy Calculations groups by their image in a quotient. It gives algorithms for basic 2. Subdirect Products operations in this representation and investigates how iteration