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1. CMB Online first

Jensen, Gerd; Pommerenke, Christian
On the structure of the Schild group in Relativity Theory
Alfred Schild has established conditions that Lorentz transformations map world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations. The present paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature we associate Lorentz transformations with matrices in $\mathrm{SL}(2,\mathbb{C})$. We consider the lattice of subgroups of the group originated in Schild's paper and obtain generating sets for the full group and its subgroups.

Keywords:Lorentz transformation, integer lattice, Gaussian integers, Schild group, subgroup
Categories:22E43, 20H99, 83A05

2. CMB Online first

Shravan Kumar, N.
Invariant means on a class of von Neumann Algebras related to Ultraspherical Hypergroups II
Let $K$ be an ultraspherical hypergroup associated to a locally compact group $G$ and a spherical projector $\pi$ and let $VN(K)$ denote the dual of the Fourier algebra $A(K)$ corresponding to $K.$ In this note, we show that the set of invariant means on $VN(K)$ is singleton if and only if $K$ is discrete. Here $K$ need not be second countable. We also study invariant means on the dual of the Fourier algebra $A_0(K),$ the closure of $A(K)$ in the $cb$-multiplier norm. Finally, we consider generalized translations and generalized invariant means.

Keywords:ultraspherical hypergroup, Fourier algebra, Fourier-Stieltjes algebra, invariant mean, generalized translation, generalized invariant mean
Categories:43A62, 46J10, 43A30, 20N20

3. 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

4. CMB Online first

Louder, Larsen; Wilton, Henry
Stackings and the $W$-cycles conjecture
We prove Wise's $W$-cycles conjecture: Consider a compact graph $\Gamma'$ immersing into another graph $\Gamma$. For any immersed cycle $\Lambda:S^1\to \Gamma$, we consider the map $\Lambda'$ from the circular components $\mathbb{S}$ of the pullback to $\Gamma'$. Unless $\Lambda'$ is reducible, the degree of the covering map $\mathbb{S}\to S^1$ is bounded above by minus the Euler characteristic of $\Gamma'$. As a corollary, any finitely generated subgroup of a one-relator group has finitely generated Schur multiplier.

Keywords:free groups, one-relator groups, right-orderability
Category:20F65

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 Online first

Button, Jack
Tubular free by cyclic groups act freely on CAT(0) cube complexes
We identify when a tubular group (the fundamental group of a finite graph of groups with $\mathbb{Z}^2$ vertex and $\mathbb{Z}$ edge groups) is free by cyclic and show, using Wise's equitable sets criterion, that every tubular free by cyclic group acts freely on a CAT(0) cube complex.

Keywords:CAT(0), tubular group
Categories:20F65, 20F67, 20E08

7. CMB Online first

Liu, Ye
On chromatic functors and stable partitions of graphs
The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed that two finite graphs have isomorphic chromatic functors if and only if they have the same chromatic polynomial. The key ingredient in the proof is the use of stable partitions of graphs. The latter is shown to be closely related to chromatic functors. In this note, we further investigate some interesting properties of chromatic functors associated to simple graphs using stable partitions. Our first result is the determination of the group of natural automorphisms of the chromatic functor, which is in general a larger group than the automorphism group of the graph. The second result is that the composition of the chromatic functor associated to a finite graph restricted to the category $\mathrm{FI}$ of finite sets and injections with the free functor into the category of complex vector spaces yields a consistent sequence of representations of symmetric groups which is representation stable in the sense of Church-Farb.

Keywords:chromatic functor, stable partition, representation stability
Categories:05C15, 20C30

8. CMB Online first

Christ, Michael; Rieffel, Marc A.
Nilpotent group C*-algebras as compact quantum metric spaces
Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$. Following Connes, $M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a ``compact quantum metric space''). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

Keywords:group C*-algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth
Categories:46L87, 20F65, 22D15, 53C23, 58B34

9. CMB 2016 (vol 59 pp. 682)

Carlson, Jon F.; Chebolu, Sunil K.; Mináč, Ján
Ghosts and Strong Ghosts in the Stable Category
Suppose that $G$ is a finite group and $k$ is a field of characteristic $p\gt 0$. A ghost map is a map in the stable category of finitely generated $kG$-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow $p$-subgroup of $G$ is cyclic of order 2 or 3. In this paper we introduce and study variations of ghost maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd's generating hypothesis and show that ghosts can be detected on a finite range of degrees of Tate cohomology. We also consider maps which mimic ghosts in high degrees.

Keywords:Tate cohomology, ghost maps, stable module category, almost split sequence, periodic cohomology
Categories:20C20, 20J06, 55P42

10. CMB Online first

Ghaani Farashahi, Arash
Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups
This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be the normalized $G$-invariant measure on $G/H$ associated to the Weil's formula. Then, we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space $L^2(G/H,\mu)$.

Keywords:compact group, homogeneous space, dual space, Plancherel (trace) formula
Categories:20G05, 43A85, 43A32, 43A40

11. CMB 2016 (vol 59 pp. 617)

Nakashima, Norihiro; Terao, Hiroaki; Tsujie, Shuhei
Canonical Systems of Basic Invariants for Unitary Reflection Groups
It has been known that there exists a canonical system for every finite real reflection group. The first and the third authors obtained an explicit formula for a canonical system in the previous paper. In this article, we first define canonical systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a canonical system for every unitary reflection group.

Keywords:basic invariant, invariant theory, finite unitary reflection group
Categories:13A50, 20F55

12. CMB 2016 (vol 59 pp. 824)

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

13. 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

14. CMB 2016 (vol 59 pp. 244)

Cao, Wensheng; Huang, Xiaolin
A Note on Quaternionic Hyperbolic Ideal Triangle Groups
In this paper, the quaternionic hyperbolic ideal triangle groups are parameterized by a real one-parameter family $\{\phi_s: s\in \mathbb{R}\}$. The indexing parameter $s$ is the tangent of the quaternionic angular invariant of a triple of points in $\partial \mathbf{H}_{\mathbb{h}}^2 $ forming this ideal triangle. We show that if $s \gt \sqrt{125/3}$ then $\phi_s$ is not a discrete embedding, and if $s \leq \sqrt{35}$ then $\phi_s$ is a discrete embedding.

Keywords:quaternionic inversion, ideal triangle group, quaternionic Cartan angular invariant
Categories:20F67, 22E40, 30F40

15. 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

16. 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

17. CMB 2015 (vol 59 pp. 123)

Jensen, Gerd; Pommerenke, Christian
Discrete Space-time and Lorentz Transformations
Alfred Schild has established conditions that Lorentz transformations map world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. The problem was dealt with in the context of tensor and spinor calculus. Due to Schild's number-theoretic arguments, the subject is also interesting when isolated from its physical background. The paper of Schild is not easy to understand. Therefore we first present a streamlined version of his proof which is based on the use of null vectors. Then we present a purely algebraic proof that is somewhat shorter. Both proofs rely on the properties of Gaussian integers.

Keywords:Lorentz transformation, integer lattice, Gaussian integers
Categories:22E43, 20H99, 83A05

18. CMB 2015 (vol 58 pp. 799)

Kong, Qingjun; Guo, Xiuyun
On $s$-semipermutable or $s$-quasinormally Embedded Subgroups of Finite Groups
Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is said to be $s$-semipermutable in $G$ if $HG_{p}=G_{p}H$ for any Sylow $p$-subgroup $G_{p}$ of $G$ with $(p,|H|)=1$; $H$ is said to be $s$-quasinormally embedded in $G$ if for each prime $p$ dividing the order of $H$, a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-quasinormal subgroup of $G$. We fix in every non-cyclic Sylow subgroup $P$ of $G$ some subgroup $D$ satisfying $1\lt |D|\lt |P|$ and study the structure of $G$ under the assumption that every subgroup $H$ of $P$ with $|H|=|D|$ is either $s$-semipermutable or $s$-quasinormally embedded in $G$. Some recent results are generalized and unified.

Keywords:$s$-semipermutable subgroup, $s$-quasinormally embedded subgroup, saturated formation.
Categories:20D10, 20D20

19. CMB 2015 (vol 59 pp. 36)

Donovan, Diane M.; Griggs, Terry S.; McCourt, Thomas A.; Opršal, Jakub; Stanovský, David
Distributive and Anti-distributive Mendelsohn Triple Systems
We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for as many combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively.

Keywords:Mendelsohn triple system, quasigroup, distributive, Moufang loop, Loeschian numbers
Categories:20N05, 05B07

20. CMB 2015 (vol 58 pp. 538)

Li, Lili; Chen, Guiyun
Minimal Non Self Dual Groups
A group $G$ is self dual if every subgroup of $G$ is isomorphic to a quotient of $G$ and every quotient of $G$ is isomorphic to a subgroup of $G$. It is minimal non-self dual if every proper subgroup of $G$ is self dual but $G$ is not self dual. In this paper, the structure of minimal non-self dual groups is determined.

Keywords:minimal non-self dual group, finite group, metacyclic group, metabelian group
Category:20D15

21. 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

22. CMB 2015 (vol 58 pp. 363)

Sharma, R. K.; Sidana, Swati
Finite Semisimple Loop Algebras of Indecomposable $RA$ Loops
There are at the most seven classes of finite indecomposable $RA$ loops upto isomorphism. In this paper, we completely characterize the structure of the unit loop of loop algebras of these seven classes of loops over finite fields of characteristic greater than $2$.

Keywords:unit loop, loop algebra, indecomposable $RA$ loops
Categories:20N05, 17D05

23. CMB 2015 (vol 58 pp. 233)

Bergen, Jeffrey
Affine Actions of $U_q(sl(2))$ on Polynomial Rings
We classify the affine actions of $U_q(sl(2))$ on commutative polynomial rings in $m \ge 1$ variables. We show that, up to scalar multiplication, there are two possible actions. In addition, for each action, the subring of invariants is a polynomial ring in either $m$ or $m-1$ variables, depending upon whether $q$ is or is not a root of $1$.

Keywords:skew derivation, quantum group, invariants
Categories:16T20, 17B37, 20G42

24. CMB 2014 (vol 58 pp. 105)

Hossein-Zadeh, Samaneh; Iranmanesh, Ali; Hosseinzadeh, Mohammad Ali; Lewis, Mark L.
On Graphs Associated with Character Degrees and Conjugacy Class Sizes of Direct Products of Finite Groups
The prime vertex graph, $\Delta (X)$, and the common divisor graph, $\Gamma (X)$, are two graphs that have been defined on a set of positive integers $X$. Some properties of these graphs have been studied in the cases where either $X$ is the set of character degrees of a group or $X$ is the set of conjugacy class sizes of a group. In this paper, we gather some results on these graphs arising in the context of direct product of two groups.

Keywords:prime vertex graph, common divisor graph, character degree, class sizes, graph operation
Categories:20E45, 05C25, 05C76

25. 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
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