Chain Of Subgroups Research Articles

Coupling and recoupling coefficients for Wigner’s U(4) supermultiplet symmetry

A novel procedure for evaluating Wigner coupling coefficients and Racah recoupling coefficients for U(4) in two group–subgroup chains is presented. The canonical U(4)⊃U(3)⊃U(2)⊃U(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rm U(4) \\supset U(3)\\supset U(2)\\supset U(1)$$\\end{document} coupling and recoupling coefficients are applicable to any system that possesses U(4) symmetry, while the physical U(4)⊃SUS(2)⊗SUT(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm{U(4)} \\supset \ extrm{SU}_{\\mathrm{S}}(2) \\otimes \ extrm{SU}_{\\mathrm{T}}(2)$$\\end{document} coupling coefficients are more specific to nuclear structure studies that utilize Wigner’s supermultiplet symmetry concept. The procedure that is proposed sidesteps the use of binomial coefficients and alternating sum series and consequently enables fast and accurate computation of any and all U(4)-underpinned features. The inner multiplicity of a (S, T) pair within a single U(4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm U(4)$$\\end{document} irreducible representation is obtained from the dimension of the null space of the SU(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm{SU(2)}$$\\end{document} raising generators, while the resolution for the outer multiplicity follows from the work of Alex et al. on U(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm U(N)$$\\end{document}. It is anticipated that a C++ library will ultimately be available for determining generic coupling and recoupling coefficients associated with both the canonical and the physical group–subgroup chains of U(4).

New procedure for evaluation of U(3) coupling and recoupling coefficients

A simple method to calculate Wigner coupling coefficients and Racah recoupling coefficients for U(3) in two group–subgroup chains is presented. While the canonical U(3)⊃U(2)⊃U(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rm U(3)\\supset U(2)\\supset U(1)$$\\end{document} coupling and recoupling coefficients are applicable to any system that respects U(3) symmetry, the U(3)⊃SO(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rm U(3) \\supset SO(3)$$\\end{document} coupling coefficients are more specific to nuclear structure studies. This new procedure precludes the use of binomial coefficients and alternating sums which were used in the 1973 formulation of Draayer and Akiyama, and in so doing provides a faster and more accurate determination of any and all required results. The resolution of the outer multiplicity is based on the null space concept of the U(3) generators proposed by Alex et al., whereas the inner multiplicity in the angular momentum subgroup chain is obtained from the dimension of the null space of the SO(3) raising operator. It is anticipated that a C++ library will ultimately be available for determining generic coupling and recoupling coefficients associated with both the canonical and the physical group–subgroup chains of U(3).

A local approach to stability groups

In this short note we prove a local version of Philip Hall’s theorem on the nilpotency of the stability group of a chain of subgroups by only using elementary commutator calculus (Hall’s theorem is a direct consequence of our result). This provides a new way of dealing with stability groups.

Irredundant bases for the symmetric group

AbstractAn irredundant base of a group acting faithfully on a finite set is a sequence of points in that produces a strictly descending chain of pointwise stabiliser subgroups in , terminating at the trivial subgroup. Suppose that is or acting primitively on , and that the point stabiliser is primitive in its natural action on points. We prove that the maximum size of an irredundant base of is , and in most cases . We also show that these bounds are best possible.

Steep uncountable groups

AbstractWe produce a simple group of cardinality which is Artinian (every strictly descending chain of subgroups is finite), satisfies a Burnside law and such that for each uncountable subset there exists a natural number for which every element of may be expressed as a product of length at most of elements in . In particular this group is Jónsson (every proper subgroup is of strictly smaller cardinality) and strongly bounded (every abstract action on a metric space has bounded orbits); this is the first example of an uncountable group having both of these properties, which is constructed without using the continuum hypothesis. The group can also be made so that all subgroups are simple and all non‐trivial subgroups are malnormal in .

Concentration of invariant means and dynamics of chain stabilizers in continuous geometries

We prove a concentration inequality for invariant means on topological groups, namely for such adapted to a chain of amenable topological subgroups. The result is based on an application of Azuma’s martingale inequality and provides a method for establishing extreme amenability. Building on this technique, we exhibit new examples of extremely amenable groups arising from von Neumann’s continuous geometries. Along the way, we also answer a question by Pestov on dynamical concentration in direct products of amenable topological groups.

On σ-Residuals of Subgroups of Finite Soluble Groups

Let σ={σi:i∈I} be a partition of the set of all prime numbers. A subgroup H of a finite group G is said to be σ-subnormal in G if H can be joined to G by a chain of subgroups H=H0⊆H1⊆⋯⊆Hn=G where, for every j=1,⋯,n, Hj−1 is normal in Hj or Hj/CoreHj(Hj−1) is a σi-group for some i∈I. Let B be a subgroup of a soluble group G normalising the Nσ-residual of every non-σ-subnormal subgroup of G, where Nσ is the saturated formation of all σ-nilpotent groups. We show that B normalises the Nσ-residual of every subgroup of G if G does not have a section that is σ-residually critical.

Finite groups with absolutely $\mathfrak{F}$-subnormal maximal subgroups

MSC: 20D10; 20E28 DOI: 10.21538/0134-4889-2023-29-1-254-258 This work was supported by the Ministry of Education of the Republic of Belarus (Grant number 20211467). A subgroup $M$ of a group $G$ is an $n$-maximal subgroups of $G$ if there is a subgroup chain $M=M_n\leq M_{n-1}\leq \ldots \leq M_1\leq M_0=G$ such that $M_{i+1}$ is a maximal subgroup of $M_i$. We establish a

Some notes on σ-soluble groups and σ-subnormality

Let G be a finite group and be a partition of the set of all primes , that is, and for all . The natural numbers n and m are called σ-coprime if . The group G is said to be: σ-primary if G is a σi -group for some ; σ-soluble if either G = 1 or every chief factor of G is σ-primary. A subgroup H of G is called σ-subnormal in G if there is a subgroup chain such that either is normal in Hi or is σ-primary for all . In this paper, we show that G is σ-soluble provided G satisfies the following conditions: (1) , where A 1, A 2, A 3 are all σ-soluble; (2) the three indices are pairwise σ-coprime. And we prove that if G is σ-soluble and A is a subgroup of G, then A is σ-subnormal in G if and only if divides for every Hall σi -subgroup B of G and all . We also state and prove a σ-nilpotency criterion for G and a characterization of the σ-Fitting subgroup of G, which are related to this observation.

A new algorithm to compute fuzzy subgroups of a finite group

<abstract><p>The enumeration of fuzzy subgroups is a complex problem. Several authors have computed results for special instances of groups. In this paper, we present a novel algorithm that is designed to enumerate the fuzzy subgroups of a finite group. This is achieved through the computation of maximal chains of subgroups. This approach is also useful for writing a program to compute the number of fuzzy subgroups. We provide further elucidation by computing the number of fuzzy subgroups of the groups $ Q_8 $ and $ D_8 $.</p></abstract>

On the Left Connected Subalgebra of the Descent Algebra of a Coxeter Group of Classical Type

A Coxeter group of classical type A_n, B_n or D_n contains a chain of subgroups of the same type. We show that intersections of conjugates of these subgroups are again of the same type, and make precise in which sense and to what extent this property is exclusive to the classical types of Coxeter groups. As the main tool for the proof, we use Solomon’s descent algebra. Using Stirling numbers, we express certain basis elements of the descent algebra as polynomials and derive explicit multiplication formulas for a commutative subalgebra of the descent algebra.

Normalizers of chains of discrete p-toral subgroups in compact Lie groups

In this paper we study the normalizer decomposition of a compact Lie group G using the information of the fusion system F of G on a maximal discrete p-toral subgroup. We prove that there is an injective map from the set of conjugacy classes of chains of F-centric, F-radical discrete p-toral subgroups to the set of conjugacy classes of chains of p-centric, p-stubborn continuous p-toral subgroups. The map is a bijection when π0(G) is a finite p-group. We also prove that the classifying space of the normalizer of a chain of discrete p-toral subgroups of G is mod p equivalent to the classifying space of the normalizer of the corresponding chain of p-toral subgroups.

On the Subgroups Lattice and Fuzzy Subgroups of Finite Groups U 6n

In this paper, we treat accounting for the number of fuzzy (normal) subgroups of finite groups . In order to do this, we use the natural equivalence relation on the set of fuzzy (normal) subgroups of , which has a consistent group theoretical foundation. In fact, the corresponding equivalence classes of fuzzy (normal) subgroups of are closely connected to the structure of (normal) subgroups lattice of and chains of subgroups of , which terminate in . In this regards, the Inclusion-Exclusion principle plays an essential role and in some situations leads to recurrence relations, whose their solutions can be easily found.

Finite groups with σ-abnormal or 𝔉-subnormal σ-primary subgroups

Let [Formula: see text] be some partition of the set of all primes [Formula: see text], [Formula: see text] a finite group and [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall[Formula: see text]-set of [Formula: see text] if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] for some [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-primary if it is a finite [Formula: see text]-group for some [Formula: see text]; [Formula: see text]-abnormal in [Formula: see text] if [Formula: see text] is not [Formula: see text]-primary whenever [Formula: see text] and [Formula: see text] is a maximal subgroup of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-subnormal if there is a chain of subgroups [Formula: see text] such that [Formula: see text] for all [Formula: see text]. This is equivalent to [Formula: see text]. In this paper, we study the structure of a finite group [Formula: see text] in which every [Formula: see text]-primary cyclic subgroups are [Formula: see text]-quasinormal or [Formula: see text]-abnormal. We also describe the structure of a finite group [Formula: see text] with self-normalizing or [Formula: see text]-subnormal [Formula: see text]-primary cyclic subgroups for a subgroup-closed saturated lattice formation [Formula: see text] containing all [Formula: see text]-nilpotent groups.

On σ-subnormality criteria in finite groups

Let σ = { σ i : i ∈ I } be a partition of the set P of all prime numbers. A subgroup H of a finite group G is called σ - subnormal in G if there is a chain of subgroups H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n = G where, for every i = 1 , … , n , H i − 1 normal in H i or H i / C o r e H i ( H i − 1 ) is a σ j -group for some j ∈ I . In the special case that σ is the partition of P into sets containing exactly one prime each, the σ -subnormality reduces to the familiar case of subnormality. In this paper some σ -subnormality criteria for subgroups of finite groups are studied.

Finite groups with given systems of generalised σ-permutable subgroups

Let σ = {σi|i ∈ I } be a partition of the set of all primes ℙ and G be a finite group. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if every member ≠1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ⌒ π(G) ≠ ∅. A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable in G if G possesses a complete Hall σ-set ℋ such that AH x = H xA for all H ∈ ℋ and all x ∈ G; σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ … ≤ At = G such that either Ai − 1 ⊴ Ai or Ai /(Ai − 1)Ai is σ-primary for all i = 1, …, t; 𝔄-normal in G if every chief factor of G between AG and AG is cyclic. We say that a subgroup H of G is: (i) partially σ-permutable in G if there are a 𝔄-normal subgroup A and a σ-permutable subgroup B of G such that H = < A, B >; (ii) (𝔄, σ)-embedded in G if there are a partially σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H. We study G assuming that some subgroups of G are partially σ-permutable or (𝔄, σ)-embedded in G. Some known results are generalised.

A chain of normalizers in the sylow $$\mathbf{2}$$-subgroups of the symmetric group on $${\mathbf{2}}^{\varvec{n}}$$ letters

On the basis of an initial interest in symmetric cryptography, in the present work we study a chain of subgroups. Starting from a Sylow $2$-subgroup of AGL(2,n), each term of the chain is defined as the normalizer of the previous one in the symmetric group on $2^n$ letters. Partial results and computational experiments lead us to conjecture that, for large values of $n$, the index of a normalizer in the consecutive one does not depend on $n$. Indeed, there is a strong evidence that the sequence of the logarithms of such indices is the one of the partial sums of the numbers of partitions into at least two distinct parts.

Generating infinite monoids of cellular automata

For a group [Formula: see text] and a set [Formula: see text], let End[Formula: see text] be the monoid of all cellular automata over [Formula: see text], and let Aut[Formula: see text] be its group of units. By establishing a characterization of surjunctive groups in terms of the monoid End[Formula: see text], we prove that the rank of End[Formula: see text] (i.e. the smallest cardinality of a generating set) is equal to the rank of Aut[Formula: see text] plus the relative rank of Aut[Formula: see text] in End[Formula: see text], and that the latter is infinite when [Formula: see text] has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when [Formula: see text] is a vector space over a field [Formula: see text], we study the monoid [Formula: see text] of all linear cellular automata over [Formula: see text] and its group of units [Formula: see text]. We show that if [Formula: see text] is an indicable group and [Formula: see text] is finite-dimensional, then [Formula: see text] is not finitely generated; however, for any finitely generated indicable group [Formula: see text], the group [Formula: see text] is finitely generated if and only if [Formula: see text] is finite.

Finite groups with K--subnormal Schmidt subgroups

Throughout this article, all groups are finite. Let be a formation of groups. A subgroup A of a finite group G is said to be K- -subnormal in G if there is a subgroup chain such that either or for all . The formation is called K-lattice if in every finite group G the set , of all K--subnormal subgroups of G, is a sublattice of the lattice of all subgroups of G. In this article, we prove that if is a hereditary K-lattice saturated formation containing all nilpotent groups and G is a finite group such that every Schmidt subgroup of G is either K--subnormal or modular in G, then the derived subgroup of G belongs to

Finite groups with σ-abnormal or σ-subnormal σ-primary subgroups

Let G be a finite group and be a partition of the set of all primes that is, and for all A set of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of is a Hall σi -subgroup of G for some i and contains exactly one Hall σi -subgroup of G for every A group is said to be σ-primary if it is a finite σi -group for some i. A subgroup A of G is said to be: σ-subnormal in G if there is a subgroup chain such that either is normal in Ai or is σ-primary for all σ-abnormal in G if is not σ-primary whenever and L is a maximal subgroup of K. In this article, we study the structure of a finite group in which σ-primary cyclic subgroups are σ-abnormal or σ-subnormal in G. We also describe the structure of a finite group G which has a complete Hall σ-set such that for any G has a subgroup A of order and A is σ-abnormal or σ-subnormal in G.