scholarly journals DENSELY ORDERED BRAID SUBGROUPS

2007 ◽  
Vol 16 (07) ◽  
pp. 869-877 ◽  
Author(s):  
ADAM CLAY ◽  
DALE ROLFSEN
Keyword(s):  
Normal Subgroup ◽  
Commutator Subgroup ◽  
Braid Groups ◽  
Burau Representation ◽  
Positive Elements

Dehornoy showed that the Artin braid groups Bn are left-orderable. This ordering is discrete, but we show that, for n > 2 the Dehornoy ordering, when restricted to certain natural subgroups, becomes a dense ordering. Among subgroups which arise are the commutator subgroup and the kernel of the Burau representation (for those n for which the kernel is nontrivial). These results follow from a characterization of least positive elements of any normal subgroup of Bn which is discretely ordered by the Dehornoy ordering.

Notes on Splitting Extensions of Groups

1968 ◽  
Vol 11 (3) ◽  
pp. 371-374 ◽  
Author(s):  
C.Y. Tang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Commutator Subgroup ◽  
Similar Type ◽  
Frattini Subgroup ◽  
Abelian Normal Subgroup ◽  
A Finite Group

In [1] Gaschütz has shown that a finite group G splits over an abelian normal subgroup N if its Frattini subgroup ϕ(G) intersects N trivially. When N is a non-abelian nilpotent normal subgroup of G the condition ϕ(G)∩ N = 1 cannot be satisfied: for if N is non-abelian then the commutator subgroup C(N) of N is non-trivial. Now N is nilpotent, whence 1 ≠ C(N)⊂ϕ(N). Since G is a finite group, therefore, by (3, theorem 7.3.17) ϕ⊂ϕ(G). It follows that ϕ(G) ∩ N ≠ 1. Thus the condition ϕ(G) ∩ N = 1 must be modified. In §1 we shall derive some similar type of conditions for G to split over N when the restriction of N being an abelian normal subgroup is removed. In § 2 we shall give a characterization of splitting extensions of N in which every subgroup splits over its intersection with N.

A Characterization of the Commutator Subgroup of a Group

1976 ◽  
Vol 19 (1) ◽  
pp. 93-94 ◽  
Author(s):  
G. Thierrin
Keyword(s):  
Normal Subgroup ◽  
Commutator Subgroup

AbstractAn element a of a semigroup S is n-potent if there exist a1, a2,..., ak∈S such that a = a1a2...ak and If S is a group, the set of n-potent elements is a normal subgroup of S and the set of 1-potent elements is the commutator subgroup of S.

scholarly journals One-relator groups and algebras related to polyhedral products

2021 ◽  
pp. 1-20
Author(s):  
Jelena Grbić ◽  
George Simmons ◽  
Marina Ilyasova ◽  
Taras Panov
Keyword(s):  
Homology Group ◽  
Homotopy Theory ◽  
Commutator Subgroup ◽  
Homology Groups ◽  
Homological Characterization ◽  
One Relator Groups ◽  
Combinatorial Condition ◽  
The Moment ◽  
Necessary And Sufficient

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

A Useful Characterization of a Normal Subgroup

1979 ◽  
Vol 52 (3) ◽  
pp. 171
Author(s):  
Francis E. Masat
Keyword(s):  
Normal Subgroup

Equivalent norms on Banach Jordan algebras

1979 ◽  
Vol 86 (2) ◽  
pp. 261-270 ◽  
Author(s):  
M. A. Youngson
Keyword(s):  
Jordan Algebra ◽  
Sufficient Conditions ◽  
Equivalent Norm ◽  
Jordan Algebras ◽  
The Self ◽  
Equivalent Norms ◽  
Positive Elements ◽  
Definition Of ◽  
Necessary And Sufficient

1. Introduction. Recently Kaplansky suggested the definition of a suitable Jordan analogue of B*-algebras, which we call J B*-algebras (see (10) and (11)). In this article, we give a characterization of those complex unital Banach Jordan algebras which are J B*-algebras in an equivalent norm. This is done by generalizing results of Bonsall ((3) and (4)) to give necessary and sufficient conditions on a real unital Banach Jordan algebra under which it is the self-adjoint part of a J B*-algebra in an equivalent norm. As a corollary we also obtain a characterization of the cones in a Banach Jordan algebra which are the set of positive elements of a J B*-algebra.

scholarly journals EXTENDING REPRESENTATIONS OF BRAID GROUPS TO THE AUTOMORPHISM GROUPS OF FREE GROUPS

2005 ◽  
Vol 14 (08) ◽  
pp. 1087-1098 ◽  
Author(s):  
VALERIJ G. BARDAKOV
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Linear Representation ◽  
Braid Groups ◽  
Burau Representation ◽  
Pure Braid Group

We construct a linear representation of the group IA (Fn) of IA-automorphisms of a free group Fn, an extension of the Gassner representation of the pure braid group Pn. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cbn contains a non-trivial kernel even if n = 2. We construct also an extension of the Burau representation to the group of conjugating automorphisms Cn. This representation is not faithful for n ≥ 2.

scholarly journals Strongly exposed points and a characterization of l1 (Γ) by the Schur property

1987 ◽  
Vol 30 (3) ◽  
pp. 397-400 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Spaces ◽  
Banach Lattice ◽  
Positive Cone ◽  
Schur Property ◽  
Positive Elements ◽  
Strongly Exposed Points ◽  
Ordered Banach Spaces ◽  
Convex Subsets ◽  
Quasi Interior

In this paper we study the existence of strongly exposed points in unbounded closed and convex subsets of the positive cone of ordered Banach spaces and we prove the following characterization for the space l1(Γ): A Banach lattice X is order-isomorphic to l1(Γ) iff X has the Schur property and X* has quasi-interior positive elements.

A Useful Characterization of a Normal Subgroup

1979 ◽  
Vol 52 (3) ◽  
pp. 171-173
Author(s):  
Francis E. Masat
Keyword(s):  
Normal Subgroup

LIE SOLVABLE GROUP ALGEBRAS OF DERIVED LENGTH THREE IN CHARACTERISTIC THREE

2012 ◽  
Vol 11 (05) ◽  
pp. 1250098 ◽  
Author(s):  
HARISH CHANDRA ◽  
MEENA SAHAI
Keyword(s):  
Solvable Group ◽  
Commutator Subgroup ◽  
Group Algebras ◽  
Derived Length ◽  
Engel Group

In this paper we provide a characterization of Lie solvable group algebras of derived length three over a field of characteristic three when G is a non-2-Engel group with abelian commutator subgroup.

Non-trivial links and plats with trivial Gassner matrices

1996 ◽  
Vol 119 (1) ◽  
pp. 43-53 ◽  
Author(s):  
Tim D. Cochran
Keyword(s):  
Normal Subgroup ◽  
Braid Group ◽  
Burau Representation ◽  
Artin Braid Group

LetBndenote the Artin braid group on ‘n-strings[ and PBnits normal subgroup consisting of all the pure braids [Bi, Mo]. These groups have been considerably scrutinized by both topologists and algebraists [BL]. One question whose answer has so far eluded us is whether or not the Gassner representationG: PBn→Mn×n(λ), into the group ofn-by-nmatrices over, is faithful (see Section 1) [Bi; ·3] [Ga]. Recently the less discriminating Burau representation B: PBn→Mn×n(Z[t±1] ) was shown to have a non-trivial kernel for each n ≥ 6 [M, LP] but these techniques have not yet yielded an element of kernel(G). This paper is a partial step in that direction.

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