PERIODIC ORBITS OF A DYNAMICAL SYSTEM RELATED TO A KNOT

Following [6] we consider a knot group G, its commutator subgroup K = [G, G], a finite group Σ and the space Hom (K, Σ) of all representations ρ : K → Σ, endowed with the weak topology. We choose a meridian x ∈ G of the knot and consider the homeomorphism σx of Hom (K, Σ) onto itself: σxρ(a) = ρ(xax-1) ∀ a ∈ K, ρ ∈ Hom (K, Σ). As proven in [5], the dynamical system ( Hom (K, Σ), σx) is a shift of finite type. In the case when Σ is abelian, Hom (K, Σ) is finite. In this paper we calculate the periods of orbits of ( Hom (K, ℤ/p), σx), where p is prime, in terms of the roots of the Alexander polynomial of the knot. In the case of two-bridge knots we give a complete description of the set of periods.

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ON CYCLES AND COVERINGS ASSOCIATED TO A KNOT

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216513500740 ◽

2013 ◽

Vol 22 (13) ◽

pp. 1350074

Author(s):

LILYA LYUBICH ◽

MIKHAIL LYUBICH

Keyword(s):

Dynamical System ◽

Abelian Group ◽

Finite Type ◽

Weak Topology ◽

Knot Theory ◽

Commutator Subgroup ◽

Finite Abelian Group ◽

Complete Classification ◽

And Shifts

Let [Formula: see text] be a knot, G be the knot group, K be its commutator subgroup, and x be a distinguished meridian. Let Σ be a finite abelian group. The dynamical system introduced by Silver and Williams in [Augmented group systems and n-knots, Math. Ann.296 (1993) 585–593; Augmented group systems and shifts of finite type, Israel J. Math.95 (1996) 231–251] consisting of the set Hom (K, Σ) of all representations ρ : K → Σ endowed with the weak topology, together with the homeomorphism [Formula: see text] is finite, i.e. it consists of several cycles. In [Periodic orbits of a dynamical system related to a knot, J. Knot Theory Ramifications20(3) (2011) 411–426] we found the lengths of these cycles for Σ = ℤ/p,p is prime, in terms of the roots of the Alexander polynomial of the knot, mod p. In this paper we generalize this result to a general abelian group Σ. This gives a complete classification of depth 2 solvable coverings over [Formula: see text].

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Notes on Splitting Extensions of Groups

Canadian Mathematical Bulletin ◽

10.4153/cmb-1968-041-3 ◽

1968 ◽

Vol 11 (3) ◽

pp. 371-374 ◽

Cited By ~ 2

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.

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On Finite Quasi-Core-p p-Groups

Symmetry ◽

10.3390/sym11091147 ◽

2019 ◽

Vol 11 (9) ◽

pp. 1147

Author(s):

Jiao Wang ◽

Xiuyun Guo

Keyword(s):

Finite Group ◽

Positive Integer ◽

Commutator Subgroup ◽

Nilpotency Class ◽

Frattini Subgroup ◽

Normal Core ◽

A Finite Group

Given a positive integer n, a finite group G is called quasi-core-n if ⟨ x ⟩ / ⟨ x ⟩ G has order at most n for any element x in G, where ⟨ x ⟩ G is the normal core of ⟨ x ⟩ in G. In this paper, we investigate the structure of finite quasi-core-p p-groups. We prove that if the nilpotency class of a quasi-core-p p-group is p + m , then the exponent of its commutator subgroup cannot exceed p m + 1 , where p is an odd prime and m is non-negative. If p = 3 , we prove that every quasi-core-3 3-group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. We also show that the Frattini subgroup of a quasi-core-2 2-group is abelian.

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A characterization of topologically completely positive entropy for shifts of finite type

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2013.18 ◽

2013 ◽

Vol 34 (6) ◽

pp. 2054-2065 ◽

Cited By ~ 1

Author(s):

RONNIE PAVLOV

Keyword(s):

Dynamical System ◽

Finite Type ◽

American Mathematical Society ◽

Equivalent Condition ◽

Positive Entropy ◽

Completely Positive ◽

Shift Of Finite Type ◽

Formula Group ◽

Zero Entropy

AbstractA topological dynamical system was defined by Blanchard [Fully Positive Topological Entropy and Topological Mixing (Symbolic Dynamics and Applications (in honor of R. L. Adler), 135). American Mathematical Society Contemporary Mathematics, Providence, RI, 1992, pp. 95–105] to have topologically completely positive entropy (or TCPE) if its only zero entropy factor is the dynamical system consisting of a single fixed point. For ${ \mathbb{Z} }^{d} $ shifts of finite type, we give a simple condition equivalent to having TCPE. We use our characterization to derive a similar equivalent condition to TCPE for the subclass of ${ \mathbb{Z} }^{d} $ group shifts, which was proved by Lind and Schmidt in the abelian case [Homoclinic points of algebraic ${ \mathbb{Z} }^{d} $-actions. J. Amer. Math. Soc. 12(4) (1999), 953–980] and by Boyle and Schraudner in the general case [${ \mathbb{Z} }^{d} $ group shifts and Bernoulli factors. Ergod. Th. & Dynam. Sys. 28(2) (2008), 367–387]. We also give an example of a ${ \mathbb{Z} }^{2} $ shift of finite type which has TCPE but is not even topologically transitive, and prove a result about block gluing ${ \mathbb{Z} }^{d} $ SFTs motivated by our characterization of TCPE.

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Periodic points and finite group actions on shifts of finite type

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700007495 ◽

1993 ◽

Vol 13 (3) ◽

pp. 485-514 ◽

Cited By ~ 2

Author(s):

Ulf-Rainer Fiebig

Keyword(s):

Finite Group ◽

Zeta Function ◽

Finite Type ◽

Ordered Set ◽

Zeta Functions ◽

Orbit Space ◽

Periodic Points ◽

Shift Of Finite Type ◽

Finite Group Actions ◽

Periodic Data

AbstractLet G be an abstract finite group. For an action α of G on a shift of finite type (SFT) S we introduce the periodic data of α, a computable finite-ordered set of complex polynomials. We show that two actions of G on possibly different SFTs are conjugate on periodic points iff their periodic data coincide. For each subgroup H of G the points fixed by α|H (the restriction of α to H) form a subsystem of S, which is of finite type. Our result shows that the zeta functions of these subsystems determine the conjugacy class (on periodic points) of α up to finitely many possibilities.The orbit space of a finite skew action on an SFT S, endowed with the homeomorphism induced by S, is shown to have a zeta function equal to the zeta function of an SFT which is a left-closing quotient of S. We show that this zeta function equals the zeta function of S iff the skew action is inert with respect to a certain power of S.Finally we consider functions of the periodic data as for example gyration numbers.

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Remarks on the commutativity of the radicals of group algebras

Glasgow Mathematical Journal ◽

10.1017/s0017089500003736 ◽

1979 ◽

Vol 20 (1) ◽

pp. 63-68 ◽

Cited By ~ 5

Author(s):

Shigeo Koshitani

Keyword(s):

Finite Group ◽

Group Algebra ◽

Sylow Subgroup ◽

Jacobson Radical ◽

Commutator Subgroup ◽

Group Algebras ◽

Arbitrary Field ◽

A Finite Group

Let K be an arbitrary field with characteristic p > 0, G a finite group of order pag′ with (p, g′) = 1, P a p-Sylow subgroup of G and G′ the commutator subgroup of G. For a ring R denote by J(R) the Jacobson radical of R and by Z(R) the centre of R. We write KG for the group algebra of G over K.

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Groups sharing some varietal properties with supersoluble groups

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700023296 ◽

1983 ◽

Vol 34 (2) ◽

pp. 265-268 ◽

Cited By ~ 3

Author(s):

Rolf Brandl

Keyword(s):

Finite Group ◽

General Result ◽

Commutator Subgroup ◽

Saturated Formation ◽

Supersoluble Group ◽

Supersoluble Groups ◽

A Finite Group ◽

Sylow Tower

AbstractIn this note a formation U is considered which can be defined by a sequence of laws which ‘almost’ hold in every finite supersoluble group. The class U contains all finite supersoluble groups and each group in U has a Sylow tower.It is shown that a finite group belongs to U if and only if all of its subgroups with nilpotent commutator subgroup are supersoluble. A more general result concerning classes of this type finally proves that U is a saturated formation.

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TOPOLOGICAL STABLE RANK OF INCLUSIONS OF UNITAL C*-ALGEBRAS

International Journal of Mathematics ◽

10.1142/s0129167x06003345 ◽

2006 ◽

Vol 17 (01) ◽

pp. 19-34 ◽

Cited By ~ 2

Author(s):

HIROYUKI OSAKA ◽

TAMOTSU TERUYA

Keyword(s):

Finite Group ◽

Tensor Product ◽

Finite Type ◽

Stable Rank ◽

C Algebras ◽

Rank One ◽

Topological Stable Rank ◽

Crossed Product Algebra ◽

A Finite Group ◽

Product Algebra

Let 1 ∈ A ⊂ B be an inclusion of C*-algebras of C*-index-finite type with depth 2. We try to compute the topological stable rank of B (= tsr (B)) when A has topological stable rank one. We show that tsr (B) ≤ 2 when A is a tsr boundedly divisible algebra, in particular, A is a C*-minimal tensor product UHF ⊗ D with tsr (D) = 1. When G is a finite group and α is an action of G on UHF, we know that a crossed product algebra UHF ⋊α G has topological stable rank less than or equal to two. These results are affirmative data to a generalization of a question by Blackadar in 1988.

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The Generation of the Lower Central Series

Canadian Journal of Mathematics ◽

10.4153/cjm-1965-041-5 ◽

1965 ◽

Vol 17 ◽

pp. 405-410 ◽

Cited By ~ 11

Author(s):

P. X. Gallagher

Keyword(s):

Finite Group ◽

Commutator Subgroup ◽

Lower Central Series ◽

Central Series ◽

A Finite Group

Let G be a finite group with commutator subgroup G′. In an earlier paper (4) it was shown that each element of G′ is a product of n commutators, if 4n ≥ |G′|. The object of this paper is to improve this result in two directions:Theorem 1a. If (n + 2)!n! > 2|G′| — 2, then each element of G′ is a product of n commutators.Theorem 1b. If G is a p-group, with |G′| = pa, and if n(n + 1) > a, then each element of G′ is a product of n commutators.

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