ENUMERATION OF WHITEHEAD GROUPS OF LOW ORDER

2002 ◽  
Vol 12 (05) ◽  
pp. 645-658
Author(s):  
MURAT ALP
Keyword(s):  
Group Theory ◽  
Programming Language ◽  
Crossed Module ◽  
Whitehead Group ◽  
Group Of Automorphisms ◽  
Isomorphism Classes ◽  
Crossed Modules ◽  
Whitehead Groups ◽  
A Share ◽  
Low Order

In this paper we describe a share package of functions for computing with finite, permutation crossed modules, cat1-groups and their morphisms, written using the [Formula: see text] group theory programming language. The category XMod of crossed modules is equivalent to the category Cat1 of cat1-groups and we include functions emulating the functors between these categories. The monoid of derivations of a crossed module [Formula: see text], and the corresponding monoid of sections of a cat1-group [Formula: see text], are constructed using the Whitehead multiplication. The Whitehead group of invertible derivations, together with the group of automorphisms of X, are used to construct the actor crossed module of X which is the automorphism object in XMod. We include a table of the 350 isomorphism classes of cat1-structures on groups of order at most 30 in [2].

scholarly journals ENUMERATION OF CAT1-GROUPS OF LOW ORDER

2000 ◽  
Vol 10 (04) ◽  
pp. 407-424 ◽  
Author(s):  
MURAT ALP ◽  
CHRISTOPHER D. WENSLEY
Keyword(s):  
Group Theory ◽  
Programming Language ◽  
Crossed Module ◽  
Whitehead Group ◽  
Group Of Automorphisms ◽  
Isomorphism Classes ◽  
Crossed Modules ◽  
A Share ◽  
Low Order

In this paper we describe a share package [Formula: see text] of functions for computing with finite, permutation crossed modules, cat1-groups and their morphisms, written using the [Formula: see text] group theory programming language. The category XMod of crossed modules is equivalent to the category Cat1 of cat1-groups and we include functions emulating the functors between these categories. The monoid of derivations of a crossed module [Formula: see text] , and the corresponding monoid of sections of a cat1-group [Formula: see text] , are constructed using the Whitehead multiplication. The Whitehead group of invertible derivations, together with the group of automorphisms of [Formula: see text] , are used to construct the actor crossed module of [Formula: see text] which is the automorphism object in XMod. We include a table of the 350 isomorphism classes of cat1-structures on groups of order at most 30.

Internal Crossed Modules

2003 ◽  
Vol 10 (1) ◽  
pp. 99-114 ◽  
Author(s):  
G. Janelidze
Keyword(s):  
Crossed Module ◽  
Crossed Modules

Abstract We introduce the notion of (pre)crossed module in a semiabelian category, and establish equivalences between internal reflexive graphs and precrossed modules, and between internal categories and crossed modules.

scholarly journals Group cohomology with coefficients in a crossed module

2010 ◽  
Vol 10 (2) ◽  
pp. 359-404 ◽  
Author(s):  
Behrang Noohi
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Geometric Approach ◽  
Group Cohomology ◽  
Theoretic Approach ◽  
Crossed Module ◽  
Explicit Approach ◽  
Crossed Modules ◽  
Cohomology Sequence

AbstractWe compare three different ways of defining group cohomology with coefficients in a crossed module: (1) explicit approach via cocycles; (2) geometric approach via gerbes; (3) group theoretic approach via butterflies. We discuss the case where the crossed module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed modules and also prove the ‘long’ exact cohomology sequence associated to a short exact sequence of crossed modules and weak morphisms.

SEMI-COMPLETE CROSSED MODULES OF LIE ALGEBRAS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250096
Author(s):  
A. AYTEKIN ◽  
J. M. CASAS ◽  
E. Ö. USLU
Keyword(s):  
Lie Algebras ◽  
Necessary Conditions ◽  
Crossed Module ◽  
Crossed Modules ◽  
Sufficient And Necessary Conditions

We investigate some sufficient and necessary conditions for (semi)-completeness of crossed modules in Lie algebras and we establish its relationships with the holomorphy of a crossed module. When we consider Lie algebras as crossed modules, then we recover the corresponding classical results for complete Lie algebras.

scholarly journals Quasirationality and prounipotent crossed modules

2019 ◽  
Vol 28 (13) ◽  
pp. 1940012
Author(s):  
A. M. Mikhovich
Keyword(s):  
Crossed Module ◽  
Defining Relation ◽  
Crossed Modules

We study quasirational (QR) presentations of (pro-[Formula: see text])groups, which contain aspherical presentations and their subpresentations, and also still mysterious pro-[Formula: see text]-groups with a single defining relation. Using schematization of QR-presentations and embedding of the rationalized module of relations into a diagram related to a certain prounipotent crossed module, we derive cohomological properties of pro-[Formula: see text]-groups with a single defining relation.

scholarly journals Normality and quotient in the category of crossed modules within the category of groups with operations

2019 ◽  
Vol 38 (7) ◽  
pp. 169-179
Author(s):  
Tunçar Şahan ◽  
Osman Mucuk
Keyword(s):  
Crossed Module ◽  
Crossed Modules

In this paper we define the notions of normal subcrossed module and quotient crossed module within groups with operations; and then give same properties of such crossed modules in groups with operations.

scholarly journals Liftings of crossed modules in the category of groups with operations

2019 ◽  
Vol 38 (7) ◽  
pp. 181-193
Author(s):  
H. Fulya Akız ◽  
Osman Mucuk ◽  
Tunçar Şahan
Keyword(s):  
Crossed Module ◽  
Crossed Modules ◽  
Internal Groupoid

In this paper we define the notion of lifting of a crossed module via the morphism in groups with operations and give some properties of this type of liftings. Further we prove that the lifting crossed modules of a certain crossed module are categorically equivalent to the internal groupoid actions on groups with operations, where the internal groupoid corresponds to the crossed module.

Whitehead groups of certain hyperbolic manifolds

1984 ◽  
Vol 95 (2) ◽  
pp. 299-308 ◽  
Author(s):  
A. J. Nicas ◽  
C. W. Stark
Keyword(s):  
Class Group ◽  
Universal Cover ◽  
Hyperbolic Manifolds ◽  
Fundamental Groups ◽  
Whitehead Group ◽  
Free Products ◽  
Projective Class ◽  
Aspherical Manifold ◽  
Hyperbolic Polyhedra ◽  
Whitehead Groups

An aspherical manifold is a connected manifold whose universal cover is contractible. It has been conjectured that the Whitehead groups Whj (π1 M) (including the projective class group, the original Whitehead group of π1M, and the higher Whitehead groups of [9]) vanish for any compact aspherical manifold M. The present paper considers this conjecture for twelve hyperbolic 3-manifolds constructed from regular hyperbolic polyhedra. Hyperbolic manifolds are of special interest in this regard since so much is known about their topology and geometry and very little is known about the algebraic K-theory of hyperbolic manifolds whose fundamental groups are not generalized free products.

Enumeration of parallel manipulators

Robotica ◽  
2009 ◽  
Vol 27 (4) ◽  
pp. 589-597 ◽  
Author(s):  
Roberto Simoni ◽  
Andrea Piga Carboni ◽  
Daniel Martins
Keyword(s):  
Group Theory ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Internal Symmetry ◽  
New Method ◽  
Chain Mechanism ◽  
End Effector ◽  
Group Of Automorphisms ◽  
Vertex Graph

SUMMARYIn this paper, we present a new method of enumeration of parallel manipulators with one end-effector. The method consists of enumerating all the manipulators possible with one end-effector that a single kinematic chain can originate. A very useful simplification for kinematic chain, mechanism and manipulator enumeration is their representation through graphs. The method is based on group theory where abstract structures are used to capture the internal symmetry of a structure in the form of automorphisms of a group. The concept used is orbits of the group of automorphisms of a colored vertex graph. The theory and some examples are presented to illustrate the method.

Cohomological Group Theory

2019 ◽  
pp. 327-378
Author(s):  
Graham Ellis
Keyword(s):  
Group Theory ◽  
Tensor Products ◽  
Hadamard Matrices ◽  
Central Extensions ◽  
Nonabelian Group ◽  
Products Of Groups ◽  
Crossed Modules ◽  
Nonabelian Tensor ◽  
Exact Sequences

This chapter introduces some of the basic ingredients of cohomological group theory and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: explicit cocycles, classification of abelian and nonabelian group extensions, crossed modules, crossed extensions, five-term exact sequences, Hopf’s formula, Bogomolov multipliers, relative central extensions, nonabelian tensor products of groups, and cocyclic Hadamard matrices.

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