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Search: All articles in the CMB digital archive with keyword groups

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1. CMB Online first

Jantzen, Jens Carsten
Maximal Weight Composition Factors for Weyl Modules
Fix an irreducible (finite) root system $R$ and a choice of positive roots. For any algebraically closed field $k$ consider the almost simple, simply connected algebraic group $G_k$ over $k$ with root system $k$. One associates to any dominant weight $\lambda$ for $R$ two $G_k$--modules with highest weight $\lambda$, the Weyl module $V (\lambda)_k$ and its simple quotient $L (\lambda)_k$. Let $\lambda$ and $\mu$ be dominant weights with $\mu \lt \lambda$ such that $\mu$ is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists $k$ such that $L (\mu)_k$ is a composition factor of $V (\lambda)_k$, and they exhibit an example in type $E_8$ where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact, one that uses a classification of the possible pairs $(\lambda, \mu)$, and another one that relies only on the classification of root systems.

Keywords:algebraic groups, represention theory
Categories:20G05, 20C20

2. CMB Online first

Clay, Adam; Desmarais, Colin; Naylor, Patrick
Testing bi-orderability of knot groups
We investigate the bi-orderability of two-bridge knot groups and the groups of knots with 12 or fewer crossings by applying recent theorems of Chiswell, Glass and Wilson. Amongst all knots with 12 or fewer crossings (of which there are 2977), previous theorems were only able to determine bi-orderability of 499 of the corresponding knot groups. With our methods we are able to deal with 191 more.

Keywords:knots, fundamental groups, orderable groups
Categories:57M25, 57M27, 06F15

3. CMB Online first

Karpenko, Nikita A.
Incompressibility of products of pseudo-homogeneous varieties
We show that the conjectural criterion of $p$-incompressibility for products of projective homogeneous varieties in terms of the factors, previously known in a few special cases only, holds in general. Actually, the proof goes through for a wider class of varieties which includes the norm varieties associated to symbols in Galois cohomology of arbitrary degree.

Keywords:algebraic groups, projective homogeneous varieties, Chow groups and motives, canonical dimension and incompressibility
Categories:20G15, 14C25

4. CMB Online first

De Carli, Laura; Samad, Gohin Shaikh
One-parameter groups of operators and discrete Hilbert transforms
We show that the discrete Hilbert transform and the discrete Kak-Hilbert transform are infinitesimal generator of one-parameter groups of operators in $\ell^2$.

Keywords:discrete Hilbert transform, groups of operators, isometries
Categories:42A45, 42A50, 41A44

5. CMB Online first

Akbari, Saieed; Miraftab, Babak; Nikandish, Reza
Co-Maximal Graphs of Subgroups of Groups
Let $H$ be a group. The co-maximal graph of subgroups of $H$, denoted by $\Gamma(H)$, is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma(H)$ if and only if $H=LK$. In this paper, we study the connectivity, diameter, clique number and vertex chromatic number of $\Gamma(H)$. For instance, we show that if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$ is connected with diameter at most $3$. Also, we characterize all finite groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma(H)$ is connected and moreover the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite.

Keywords:co-maximal graphs of subgroups of groups, diameter, nilpotent group, solvable group
Categories:05C25, 05E15, 20D10, 20D15

6. CMB 2016 (vol 59 pp. 234)

Beardon, Alan F.
Non-discrete Frieze Groups
The classification of Euclidean frieze groups into seven conjugacy classes is well known, and many articles on recreational mathematics contain frieze patterns that illustrate these classes. However, it is only possible to draw these patterns because the subgroup of translations that leave the pattern invariant is (by definition) cyclic, and hence discrete. In this paper we classify the conjugacy classes of frieze groups that contain a non-discrete subgroup of translations, and clearly these groups cannot be represented pictorially in any practical way. In addition, this discussion sheds light on why there are only seven conjugacy classes in the classical case.

Keywords:frieze groups, isometry groups
Categories:51M04, 51N30, 20E45

7. CMB 2016 (vol 59 pp. 392)

Prajapati, S. K.; Sarma, R.
Total Character of a Group $G$ with $(G,Z(G))$ as a Generalized Camina Pair
We investigate whether the total character of a finite group $G$ is a polynomial in a suitable irreducible character of $G$. When $(G,Z(G))$ is a generalized Camina pair, we show that the total character is a polynomial in a faithful irreducible character of $G$ if and only if $Z(G)$ is cyclic.

Keywords:finite groups, group characters, total characters
Category:20C15

8. CMB 2015 (vol 59 pp. 170)

Martínez-Pedroza, Eduardo
A Note on Fine Graphs and Homological Isoperimetric Inequalities
In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected $2$-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of attaching maps of $2$-cells and finitely many $2$-cells adjacent to any edge must have a fine $1$-skeleton. We provide a positive answer to this question. We revisit a homological characterization of relative hyperbolicity, and show that a group $G$ is hyperbolic relative to a collection of subgroups $\mathcal P$ if and only if $G$ acts cocompactly with finite edge stabilizers on an connected $2$-dimensional cell complex with a linear homological isoperimetric inequality and $\mathcal P$ is a collection of representatives of conjugacy classes of vertex stabilizers.

Keywords:isoperimetric functions, Dehn functions, hyperbolic groups
Categories:20F67, 05C10, 20J05, 57M60

9. CMB 2015 (vol 59 pp. 50)

Dorfmeister, Josef F.; Inoguchi, Jun-ichi; Kobayashi, Shimpei
On the Bernstein Problem in the Three-dimensional Heisenberg Group
In this note we present a simple alternative proof for the Bernstein problem in the three-dimensional Heisenberg group $\operatorname{Nil}_3$ by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch-Rosenberg differential.

Keywords:Bernstein problem, minimal graphs, Heisenberg group, loop groups, spinors
Categories:53A10, 53C42

10. CMB 2015 (vol 59 pp. 144)

Laterveer, Robert
A Brief Note Concerning Hard Lefschetz for Chow Groups
We formulate a conjectural hard Lefschetz property for Chow groups, and prove this in some special cases: roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional Griffiths group. An appendix includes related statements that follow from results of Vial.

Keywords:algebraic cycles, Chow groups, finite-dimensional motives
Categories:14C15, 14C25, 14C30

11. CMB 2015 (vol 58 pp. 497)

Edmunds, Charles C.
Constructing Double Magma on Groups Using Commutation Operations
A magma $(M,\star)$ is a nonempty set with a binary operation. A double magma $(M, \star, \bullet)$ is a nonempty set with two binary operations satisfying the interchange law, $(w \star x) \bullet (y\star z)=(w\bullet y)\star(x \bullet z)$. We call a double magma proper if the two operations are distinct and commutative if the operations are commutative. A double semigroup, first introduced by Kock, is a double magma for which both operations are associative. Given a non-trivial group $G$ we define a system of two magma $(G,\star,\bullet)$ using the commutator operations $x \star y = [x,y](=x^{-1}y^{-1}xy)$ and $x\bullet y = [y,x]$. We show that $(G,\star,\bullet)$ is a double magma if and only if $G$ satisfies the commutator laws $[x,y;x,z]=1$ and $[w,x;y,z]^{2}=1$. We note that the first law defines the class of 3-metabelian groups. If both these laws hold in $G$, the double magma is proper if and only if there exist $x_0,y_0 \in G$ for which $[x_0,y_0]^2 \not= 1$. This double magma is a double semigroup if and only if $G$ is nilpotent of class two. We construct a specific example of a proper double semigroup based on the dihedral group of order 16. In addition we comment on a similar construction for rings using Lie commutators.

Keywords:double magma, double semigroups, 3-metabelian
Categories:20E10, 20M99

12. CMB 2015 (vol 58 pp. 730)

Efrat, Ido; Matzri, Eliyahu
Vanishing of Massey Products and Brauer Groups
Let $p$ be a prime number and $F$ a field containing a root of unity of order $p$. We relate recent results on vanishing of triple Massey products in the mod-$p$ Galois cohomology of $F$, due to Hopkins, Wickelgren, Mináċ, and Tân, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property.

Keywords:Galois cohomology, Brauer groups, triple Massey products, global fields
Categories:16K50, 11R34, 12G05, 12E30

13. CMB 2015 (vol 58 pp. 241)

Botelho, Fernanda
Isometries and Hermitian Operators on Zygmund Spaces
In this paper we characterize the isometries of subspaces of the little Zygmund space. We show that the isometries of these spaces are surjective and represented as integral operators. We also show that all hermitian operators on these settings are bounded.

Keywords:Zygmund spaces, the little Zygmund space, Hermitian operators, surjective linear isometries, generators of one-parameter groups of surjective isometries
Categories:46E15, 47B15, 47B38

14. CMB 2014 (vol 58 pp. 182)

Tărnăuceanu, Marius
On Finite Groups with Dismantlable Subgroup Lattices
In this note we study the finite groups whose subgroup lattices are dismantlable.

Keywords:finite groups, subgroup lattices, dismantlable lattices, planar lattices, crowns
Categories:20D30, 20D60, 20E15

15. CMB 2014 (vol 58 pp. 3)

Alaghmandan, Mahmood
Approximate Amenability of Segal Algebras II
We prove that every proper Segal algebra of a SIN group is not approximately amenable.

Keywords:Segal algebras, approximate amenability, SIN groups, commutative Banach algebras
Categories:46H20, 43A20

16. CMB 2014 (vol 58 pp. 69)

Fulp, Ronald Owen
Correction to "Infinite Dimensional DeWitt Supergroups and Their Bodies"
The Theorem below is a correction to Theorem 3.5 in the article entitled " Infinite Dimensional DeWitt Supergroups and Their Bodies" published in Canad. Math. Bull. Vol. 57 (2) 2014 pp. 283-288. Only part (iii) of that Theorem requires correction. The proof of Theorem 3.5 in the original article failed to separate the proof of (ii) from the proof of (iii). The proof of (ii) is complete once it is established that $ad_a$ is quasi-nilpotent for each $a$ since it immediately follows that $K$ is quasi-nilpotent. The proof of (iii) is not complete in the original article. The revision appears as the proof of (iii) of the revised Theorem below.

Keywords:super groups, body of super groups, Banach Lie groups
Categories:58B25, 17B65, 81R10, 57P99

17. CMB 2014 (vol 57 pp. 708)

Brannan, Michael
Strong Asymptotic Freeness for Free Orthogonal Quantum Groups
It is known that the normalized standard generators of the free orthogonal quantum group $O_N^+$ converge in distribution to a free semicircular system as $N \to \infty$. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator norm of any non-commutative polynomial in the normalized standard generators of $O_N^+$ converges as $N \to \infty$ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well known $L^2$-$L^\infty$ norm equivalence for non-commutative polynomials in free semicircular systems.

Keywords:quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decay
Categories:46L54, 20G42, 46L65

18. CMB 2014 (vol 57 pp. 648)

Tang, Juping; Miao, Long
On the ${\mathcal F}{\Phi}$-Hypercentre of Finite Groups
Let $G$ be a finite group, $\mathcal F$ a class of groups. Then $Z_{{\mathcal F}{\Phi}}(G)$ is the ${\mathcal F}{\Phi}$-hypercentre of $G$ which is the product of all normal subgroups of $G$ whose non-Frattini $G$-chief factors are $\mathcal F$-central in $G$. A subgroup $H$ is called $\mathcal M$-supplemented in a finite group $G$, if there exists a subgroup $B$ of $G$ such that $G=HB$ and $H_1B$ is a proper subgroup of $G$ for any maximal subgroup $H_1$ of $H$. The main purpose of this paper is to prove: Let $E$ be a normal subgroup of a group $G$. Suppose that every noncyclic Sylow subgroup $P$ of $F^{*}(E)$ has a subgroup $D$ such that $1\lt |D|\lt |P|$ and every subgroup $H$ of $P$ with order $|H|=|D|$ is $\mathcal M$-supplemented in $G$, then $E\leq Z_{{\mathcal U}{\Phi}}(G)$.

Keywords:${\mathcal F}{\Phi}$-hypercentre, Sylow subgroups, $\mathcal M$-supplemented subgroups, formation
Categories:20D10, 20D20

19. CMB 2014 (vol 57 pp. 511)

Gonçalves, Daniel
Simplicity of Partial Skew Group Rings of Abelian Groups
Let $A$ be a ring with local units, $E$ a set of local units for $A$, $G$ an abelian group and $\alpha$ a partial action of $G$ by ideals of $A$ that contain local units. We show that $A\star_{\alpha} G$ is simple if and only if $A$ is $G$-simple and the center of the corner $e\delta_0 (A\star_{\alpha} G) e \delta_0$ is a field for all $e\in E$. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level.

Keywords:partial skew group rings, simple rings, partial actions, abelian groups
Categories:16S35, 37B05

20. CMB 2014 (vol 57 pp. 277)

Elkholy, A. M.; El-Latif, M. H. Abd
On Mutually $m$-permutable Product of Smooth Groups
Let $G$ be a finite group and $H$, $K$ two subgroups of G. A group $G$ is said to be a mutually m-permutable product of $H$ and $K$ if $G=HK$ and every maximal subgroup of $H$ permutes with $K$ and every maximal subgroup of $K$ permutes with $H$. In this paper, we investigate the structure of a finite group which is a mutually m-permutable product of two subgroups under the assumption that its maximal subgroups are totally smooth.

Keywords:permutable subgroups, $m$-permutable, smooth groups, subgroup lattices
Categories:20D10, 20D20, 20E15, 20F16

21. CMB 2013 (vol 57 pp. 245)

Brodskiy, N.; Dydak, J.; Lang, U.
Assouad-Nagata Dimension of Wreath Products of Groups
Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is finitely generated. We show that the Assouad-Nagata dimension $\dim_{AN}(H\wr G)$ of $H\wr G$ depends on the growth of $G$ as follows: \par If the growth of $G$ is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$, otherwise $\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$.

Keywords:Assouad-Nagata dimension, asymptotic dimension, wreath product, growth of groups
Categories:54F45, 55M10, 54C65

22. CMB 2013 (vol 57 pp. 449)

Alaghmandan, Mahmood; Choi, Yemon; Samei, Ebrahim
ZL-amenability Constants of Finite Groups with Two Character Degrees
We calculate the exact amenability constant of the centre of $\ell^1(G)$ when $G$ is one of the following classes of finite group: dihedral; extraspecial; or Frobenius with abelian complement and kernel. This is done using a formula which applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk (J. Funct. Anal. 2009).

Keywords:center of group algebras, characters, character degrees, amenability constant, Frobenius group, extraspecial groups
Categories:43A20, 20C15

23. CMB 2013 (vol 57 pp. 125)

Mlaiki, Nabil M.
Camina Triples
In this paper, we study Camina triples. Camina triples are a generalization of Camina pairs. Camina pairs were first introduced in 1978 by A .R. Camina. Camina's work was inspired by the study of Frobenius groups. We show that if $(G,N,M)$ is a Camina triple, then either $G/N$ is a $p$-group, or $M$ is abelian, or $M$ has a non-trivial nilpotent or Frobenius quotient.

Keywords:Camina triples, Camina pairs, nilpotent groups, vanishing off subgroup, irreducible characters, solvable groups
Category:20D15

24. CMB 2013 (vol 57 pp. 335)

Karassev, A.; Todorov, V.; Valov, V.
Alexandroff Manifolds and Homogeneous Continua
ny homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum is a $V^n$-continuum in the sense of Alexandroff. We also prove that any finite-dimensional homogeneous metric continuum $X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq 1$, cannot be separated by a compactum $K$ with $\check{H}^{n-1}(K;G)=0$ and $\dim_G K\leq n-1$. This provides a partial answer to a question of Kallipoliti-Papasoglu whether any two-dimensional homogeneous Peano continuum cannot be separated by arcs.

Keywords:Cantor manifold, cohomological dimension, cohomology groups, homogeneous compactum, separator, $V^n$-continuum
Categories:54F45, 54F15

25. CMB 2013 (vol 57 pp. 546)

Kalantar, Mehrdad
Compact Operators in Regular LCQ Groups
We show that a regular locally compact quantum group $\mathbb{G}$ is discrete if and only if $\mathcal{L}^{\infty}(\mathbb{G})$ contains non-zero compact operators on $\mathcal{L}^{2}(\mathbb{G})$. As a corollary we classify all discrete quantum groups among regular locally compact quantum groups $\mathbb{G}$ where $\mathcal{L}^{1}(\mathbb{G})$ has the Radon--Nikodym property.

Keywords:locally compact quantum groups, regularity, compact operators
Category:46L89
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