scholarly journals Representations of virtual braids by automorphisms and virtual knot groups

2017 ◽  
Vol 26 (01) ◽  
pp. 1750003 ◽  
Author(s):  
Valeriy G. Bardakov ◽  
Yuliya A. Mikhalchishina ◽  
Mikhail V. Neshchadim
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Free Product ◽  
Free Group ◽  
Braid Group ◽  
Free Abelian Group ◽  
Virtual Link ◽  
Virtual Knot ◽  
Link Group

In the present paper, a new representation of the virtual braid group [Formula: see text] into the automorphism group of free product of the free group and free abelian group is constructed. This representation generalizes the previously constructed ones. The fact that the previously known representations are not faithful for [Formula: see text] is verified. Using representations of [Formula: see text], a virtual link group is defined. Also representations of the welded braid group [Formula: see text] are constructed and the welded link group is defined.

scholarly journals Planar pure braids on six strands

2020 ◽  
Vol 29 (01) ◽  
pp. 1950097
Author(s):  
Jacob Mostovoy ◽  
Christopher Roque-Márquez
Keyword(s):  
Abelian Group ◽  
Fundamental Group ◽  
Free Product ◽  
Free Group ◽  
Configuration Space ◽  
Braid Group ◽  
Free Abelian Group ◽  
Braid Groups ◽  
Pure Braid Group

The group of planar (or flat) pure braids on [Formula: see text] strands, also known as the pure twin group, is the fundamental group of the configuration space [Formula: see text] of [Formula: see text] labeled points in [Formula: see text] no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

scholarly journals FREE GROUPS AND AUTOMORPHISM GROUPS OF INFINITE STRUCTURES

2014 ◽  
Vol 2 ◽  
Author(s):  
PHILIPP LÜCKE ◽  
SAHARON SHELAH
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Set Theory ◽  
Free Group ◽  
Automorphism Groups ◽  
Free Abelian Group ◽  
Free Groups ◽  
Large Cardinals ◽  
Free Objects ◽  
Models Of Set Theory

AbstractGiven a cardinal $\lambda $ with $\lambda =\lambda ^{\aleph _0}$, we show that there is a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. In the proof of this statement, we develop general techniques that enable us to realize certain groups as the automorphism group of structures of a given cardinality. They allow us to show that analogues of this result hold for free objects in various varieties of groups. For example, the free abelian group of rank $2^\lambda $ is the automorphism group of a field of cardinality $\lambda $ whenever $\lambda $ is a cardinal with $\lambda =\lambda ^{\aleph _0}$. Moreover, we apply these techniques to show that consistently the assumption that $\lambda =\lambda ^{\aleph _0}$ is not necessary for the existence of a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. Finally, we use them to prove that the existence of a cardinal $\lambda $ of uncountable cofinality with the property that there is no field of cardinality $\lambda $ whose automorphism group is a free group of rank greater than $\lambda $ implies the existence of large cardinals in certain inner models of set theory.

scholarly journals Virtual braids and virtual curve diagrams

2015 ◽  
Vol 24 (13) ◽  
pp. 1541001 ◽  
Author(s):  
Oleg Chterental
Keyword(s):  
Automorphism Group ◽  
Braid Group ◽  
Knot Theory ◽  
Class Group ◽  
Mapping Class ◽  
Virtual Knot ◽  
Injective Homomorphism ◽  
Virtual Links ◽  
Upper Half Plane ◽  
Virtual Knot Theory

There is a well-known injective homomorphism [Formula: see text] from the classical braid group [Formula: see text] into the automorphism group of the free group [Formula: see text], first described by Artin [Theory of Braids, Ann. Math. (2) 48(1) (1947) 101–126]. This homomorphism induces an action of [Formula: see text] on [Formula: see text] that can be recovered by considering the braid group as the mapping class group of [Formula: see text] (an upper half plane with [Formula: see text] punctures) acting naturally on the fundamental group of [Formula: see text]. Kauffman introduced virtual links [Virtual knot theory, European J. Combin. 20 (1999) 663–691] as an extension of the classical notion of a link in [Formula: see text]. There is a corresponding notion of a virtual braid, and the set of virtual braids on [Formula: see text] strands forms a group [Formula: see text]. In this paper, we will generalize the Artin action to virtual braids. We will define a set, [Formula: see text], of “virtual curve diagrams” and define an action of [Formula: see text] on [Formula: see text]. Then, we will show that, as in Artin’s case, the action is faithful. This provides a combinatorial solution to the word problem in [Formula: see text]. In the papers [V. G. Bardakov, Virtual and welded links and their invariants, Siberian Electron. Math. Rep. 21 (2005) 196–199; V. O. Manturov, On recognition of virtual braids, Zap. Nauch. Sem. POMI 299 (2003) 267–286], an extension [Formula: see text] of the Artin homomorphism was introduced, and the question of its injectivity was raised. We find that [Formula: see text] is not injective by exhibiting a non-trivial virtual braid in the kernel when [Formula: see text].

scholarly journals Maximal normal subgroups of the integral linear group of countable degree

1976 ◽  
Vol 15 (3) ◽  
pp. 439-451 ◽  
Author(s):  
R.G. Burns ◽  
I.H. Farouqi
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Normal Subgroup ◽  
Vector Space ◽  
Free Abelian Group ◽  
Normal Structure ◽  
Normal Subgroups ◽  
Finite Degree ◽  
Infinite Rank ◽  
Countably Infinite

This paper continues the second author's investigation of the normal structure of the automorphism group г of a free abelian group of countably infinite rank. It is shown firstly that, in contrast with the case of finite degree, for each prime p every linear transformation of the vector space of countably infinite dimension over Zp, the field of p elements, is induced by an element of г Since by a result of Alex Rosenberg GL(אo, Zp ) has a (unique) maximal normal subgroup, it then follows that г has maximal normal subgroups, one for each prime.

scholarly journals Extensions of group retractions

1980 ◽  
Vol 3 (4) ◽  
pp. 719-730 ◽  
Author(s):  
Richard D. Byrd ◽  
Justin T. Lloyd ◽  
Roberto A. Mena ◽  
J. Roger Teller
Keyword(s):  
Abelian Group ◽  
Free Group ◽  
Free Abelian Group ◽  
Torsion Free Group ◽  
Torsion Free Abelian Group ◽  
Necessary And Sufficient ◽  
Torsion Free

In this paper a condition, which is necessary and sufficient, is determined when a retraction of a subgroupHof a torsion-free groupGcan be extended to a retraction ofG. It is also shown that each retraction of a torsion-free abelian group can be uniquely extended to a retraction of its divisible closure.

scholarly journals A NOTE ON NIELSEN EQUIVALENCE IN FINITELY GENERATED ABELIAN GROUPS

2011 ◽  
Vol 84 (1) ◽  
pp. 127-136 ◽  
Author(s):  
DANIEL OANCEA
Keyword(s):  
Abelian Group ◽  
Equivalence Class ◽  
Free Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Arbitrary Rank ◽  
Generating Sets

AbstractNielsen transformations determine the automorphisms of a free group of rank n, and also of a free abelian group of rank n, and furthermore the generating n-tuples of such groups form a single Nielsen equivalence class. For an arbitrary rank n group, the generating n-tuples may fall into several Nielsen classes. Diaconis and Graham [‘The graph of generating sets of an abelian group’, Colloq. Math.80 (1999), 31–38] determined the Nielsen classes for finite abelian groups. We extend their result to the case of infinite abelian groups.

Automorphisms of Free Groups

2017 ◽  
Author(s):  
Matt Clay
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Free Group ◽  
Free Groups ◽  
Research Projects ◽  
Frobenius Theorem ◽  
Automorphisms Of Free Groups ◽  
Train Tracks

This chapter discusses the automorphisms of free groups. Every group is the collection of symmetries of some object, namely, its Cayley graph. A symmetry of a group is called an automorphism; it is merely an isomorphism of the group to itself. The collection of all of the automorphisms is also a group too, known as the automorphism group and denoted by Aut (G). The chapter considers basic examples of groups to illustrate what an automorphism is, with a focus on the automorphisms of the symmetric group on three elements and of the free abelian group. It also examines the dynamics of an automorphism of a free group and concludes with a description of train tracks, a topological model for the free group, and the Perron–Frobenius theorem. Exercises and research projects are included.

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