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 worldvectors $(ct,x,y,z)$ 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
 Józiak, Paweł

Remarks on Hopf images and quantum permutation groups $S_n^+$
Motivated by a question of A. Skalski and P.M. SoÅtan (2016)
about inner faithfulness of the S. Curran's map of extending
a quantum increasing sequence to a quantum permutation, we revisit
the results and techniques of T. Banica and J. Bichon (2009)
and study some grouptheoretic properties of the quantum permutation
group on $4$ points. This enables us not only to answer the aforementioned
question in positive in case $n=4, k=2$, but also to classify
the automorphisms of $S_4^+$, describe all the embeddings $O_{1}(2)\subset
S_4^+$ and show that all the copies of $O_{1}(2)$ inside $S_4^+$
are conjugate. We then use these results to show that the converse
to the criterion we applied to answer the aforementioned question
is not valid.
Keywords:Hopf image, quantum permutation group, compact quantum group Categories:20G42, 81R50, 46L89, 16W35 

3. 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
onerelator group has finitely generated Schur multiplier.
Keywords:free groups, onerelator groups, rightorderability Category:20F65 

4. CMB Online first
 Motegi, Kimihiko; Teragaito, Masakazu

Generalized torsion elements and biorderability of 3manifold groups
It is known that a biorderable group has no generalized torsion
element,
but the converse does not hold in general.
We conjecture that the converse holds for the fundamental groups
of $3$manifolds,
and verify the conjecture for nonhyperbolic, geometric $3$manifolds.
We also confirm the conjecture for some infinite families of
closed hyperbolic $3$manifolds.
In the course of the proof,
we prove that each standard generator of the Fibonacci group
$F(2, m)$ ($m \gt 2$) is a generalized torsion element.
Keywords:generalized torsion element, biordering, 3manifold group Categories:57M25, 57M05, 06F15, 20F05 

5. CMB 2016 (vol 60 pp. 54)
 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 

6. 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, FourierStieltjes algebra, invariant mean, generalized translation, generalized invariant mean Categories:43A62, 46J10, 43A30, 20N20 

7. 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 

8. CMB 2016 (vol 60 pp. 12)
 Akbari, Saieed; Miraftab, Babak; Nikandish, Reza

Comaximal Graphs of Subgroups of Groups
Let $H$ be a group. The comaximal graph of subgroups
of $H$, denoted by $\Gamma(H)$, is a
graph whose vertices are nontrivial 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 comaximal 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
comaximal graph of a general linear group over an algebraically
closed field is zero or infinite.
Keywords:comaximal graphs of subgroups of groups, diameter, nilpotent group, solvable group Categories:05C25, 05E15, 20D10, 20D15 

9. CMB 2016 (vol 60 pp. 154)
 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 ChurchFarb.
Keywords:chromatic functor, stable partition, representation stability Categories:05C15, 20C30 

10. CMB 2016 (vol 60 pp. 77)
 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 wordlength functions
on finitely generated nilpotentbyfinite groups.
Keywords:group C*algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth Categories:46L87, 20F65, 22D15, 53C23, 58B34 

11. 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 

12. CMB 2016 (vol 60 pp. 111)
 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 

13. 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 

14. CMB 2016 (vol 59 pp. 824)
 Karpenko, Nikita A.

Incompressibility of Products of Pseudohomogeneous 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 

15. CMB 2016 (vol 59 pp. 234)
 Beardon, Alan F.

Nondiscrete 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 nondiscrete 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 

16. 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 oneparameter
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 

17. CMB 2016 (vol 59 pp. 392)
18. CMB 2015 (vol 59 pp. 170)
 MartínezPedroza, 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 

19. CMB 2015 (vol 59 pp. 123)
 Jensen, Gerd; Pommerenke, Christian

Discrete Spacetime and Lorentz Transformations
Alfred Schild has established conditions
that Lorentz transformations map worldvectors $(ct,x,y,z)$ 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 numbertheoretic 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 

20. 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 noncyclic 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 

21. CMB 2015 (vol 59 pp. 36)
 Donovan, Diane M.; Griggs, Terry S.; McCourt, Thomas A.; Opršal, Jakub; Stanovský, David

Distributive and Antidistributive 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 antimitre
Steiner triple systems respectively.
Keywords:Mendelsohn triple system, quasigroup, distributive, Moufang loop, Loeschian numbers Categories:20N05, 05B07 

22. 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 nonself dual if every
proper subgroup of $G$
is self dual but $G$ is not self dual. In this paper, the structure
of minimal nonself dual groups is determined.
Keywords:minimal nonself dual group, finite group, metacyclic group, metabelian group Category:20D15 

23. 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 nontrivial 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 3metabelian
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, 3metabelian Categories:20E10, 20M99 

24. CMB 2015 (vol 58 pp. 363)
25. 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 $m1$ variables,
depending upon whether $q$ is or is not a root of $1$.
Keywords:skew derivation, quantum group, invariants Categories:16T20, 17B37, 20G42 
