26. 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 

27. CMB 2014 (vol 58 pp. 105)
 HosseinZadeh, 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 

28. CMB 2014 (vol 58 pp. 182)
29. CMB 2014 (vol 58 pp. 196)
30. CMB 2014 (vol 57 pp. 884)
 Xu, Yong; Zhang, Xinjian

$m$embedded Subgroups and $p$nilpotency of Finite Groups
Let $A$ be a subgroup of a finite group $G$ and $\Sigma : G_0\leq
G_1\leq\cdots \leq G_n$ some subgroup series of $G$. Suppose that
for each pair $(K,H)$ such that $K$ is a maximal subgroup of $H$ and
$G_{i1}\leq K \lt H\leq G_i$, for some $i$, either $A\cap H = A\cap K$
or $AH = AK$. Then $A$ is said to be $\Sigma$embedded in $G$; $A$
is said to be $m$embedded in $G$ if $G$ has a subnormal subgroup
$T$ and a $\{1\leq G\}$embedded subgroup $C$ in $G$ such that $G =
AT$ and $T\cap A\leq C\leq A$. In this article, some sufficient
conditions for a finite group $G$ to be $p$nilpotent are given
whenever all subgroups with order $p^{k}$ of a Sylow $p$subgroup of
$G$ are $m$embedded for a given positive integer $k$.
Keywords:finite group, $p$nilpotent group, $m$embedded subgroup Categories:20D10, 20D15 

31. CMB 2014 (vol 57 pp. 621)
 Petrich, Mario

Combinatorially Factorizable Cryptic Inverse Semigroups
An inverse semigroup $S$ is combinatorially factorizable if $S=TG$
where $T$ is a combinatorial (i.e., $\mathcal{H}$ is the equality
relation) inverse subsemigroup of $S$ and $G$ is a subgroup of $S$.
This concept was introduced and studied by Mills, especially in the
case when $S$ is cryptic (i.e., $\mathcal{H}$ is a congruence on
$S$). Her approach is mainly analytical considering subsemigroups of
a cryptic inverse semigroup.
We start with a combinatorial inverse monoid and a factorizable
Clifford monoid and from an action of the former on the latter
construct the semigroups in the title. As a special case, we
consider semigroups which are direct products of a combinatorial
inverse monoid and a group.
Keywords:inverse semigroup, cryptic semigroup Category:20M18 

32. 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
noncommutative polynomial in the normalized standard generators of
$O_N^+$ converges as $N \to \infty$ to the operator norm of the
corresponding noncommutative 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 matrixcoefficient version of our strong
convergence theorem, and we recover a well known $L^2$$L^\infty$ norm
equivalence for noncommutative polynomials in free semicircular
systems.
Keywords:quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decay Categories:46L54, 20G42, 46L65 

33. 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
nonFrattini $G$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 

34. CMB 2014 (vol 57 pp. 277)
 Elkholy, A. M.; ElLatif, 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 mpermutable 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 mpermutable 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 

35. CMB 2014 (vol 57 pp. 390)
 Morita, Jun; Rémy, Bertrand

Simplicity of Some Twin Tree Automorphism Groups with Trivial Commutation Relations
We prove simplicity for incomplete rank 2 KacMoody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs.
We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite).
Nevertheless we use the fact that the latter groups are just infinite
(modulo center).
Keywords:KacMoody group, twin tree, simplicity, root system, building Categories:20G44, 20E42, 51E24 

36. CMB 2014 (vol 57 pp. 231)
 Bagherian, J.

On the Multiplicities of Characters in Table Algebras
In this paper we show that every module of a table algebra
can be considered as a faithful module of some quotient table
algebra.
Also we prove that every faithful module of a table algebra
determines a closed subset which is a cyclic group.
As a main result we give some information about multiplicities
of characters in table algebras.
Keywords:table algebra, faithful module, multiplicity of character Categories:20C99, 16G30 

37. CMB 2013 (vol 57 pp. 506)
 Galindo, César

On Braided and Ribbon Unitary Fusion Categories
We prove that every braiding over a unitary fusion category is
unitary and every unitary braided fusion category admits a unique
unitary ribbon structure.
Keywords:fusion categories, braided categories, modular categories Categories:20F36, 16W30, 18D10 

38. CMB 2013 (vol 57 pp. 449)
 Alaghmandan, Mahmood; Choi, Yemon; Samei, Ebrahim

ZLamenability 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 

39. 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 nontrivial nilpotent or
Frobenius quotient.
Keywords:Camina triples, Camina pairs, nilpotent groups, vanishing off subgroup, irreducible characters, solvable groups Category:20D15 

40. CMB 2013 (vol 57 pp. 9)
41. CMB 2013 (vol 56 pp. 795)
 MacDonald, Mark L.

Upper Bounds for the Essential Dimension of $E_7$
This paper gives a new upper bound for the essential dimension and the
essential 2dimension of the split simply connected group of type
$E_7$ over a field of characteristic not 2 or 3. In particular,
$\operatorname{ed}(E_7) \leq 29$, and $\operatorname{ed}(E_7;2) \leq 27$.
Keywords:$E_7$, essential dimension, stabilizer in general position Categories:20G15, 20G41 

42. CMB 2012 (vol 57 pp. 326)
 Ivanov, S. V.; Mikhailov, Roman

On Zerodivisors in Group Rings of Groups with Torsion
Nontrivial pairs of zerodivisors in group rings are
introduced and discussed. A problem on the existence of nontrivial
pairs of zerodivisors in group rings of free Burnside groups of odd
exponent $n \gg 1$ is solved in the affirmative. Nontrivial pairs of
zerodivisors are also found in group rings of free products of groups
with torsion.
Keywords:Burnside groups, free products of groups, group rings, zerodivisors Categories:20C07, 20E06, 20F05, , 20F50 

43. CMB 2012 (vol 57 pp. 303)
44. CMB 2012 (vol 56 pp. 881)
45. CMB 2012 (vol 57 pp. 97)
 Levy, Jason

Rationality and the JordanGattiViniberghi decomposition
We verify
our earlier conjecture
and use it to prove that the
semisimple parts of the rational JordanKacVinberg decompositions of
a rational vector all lie in a single rational orbit.
Keywords:reductive group, $G$module, Jordan decomposition, orbit closure, rationality Categories:20G15, 14L24 

46. CMB 2012 (vol 57 pp. 424)
 Sołtan, Piotr M.; Viselter, Ami

A Note on Amenability of Locally Compact Quantum Groups
In this short note we introduce a notion called ``quantum injectivity''
of locally compact quantum groups, and prove that it is equivalent
to amenability of the dual. Particularly, this provides a new characterization
of amenability of locally compact groups.
Keywords:amenability, conditional expectation, injectivity, locally compact quantum group, quantum injectivity Categories:20G42, 22D25, 46L89 

47. CMB 2012 (vol 57 pp. 132)
 Mubeena, T.; Sankaran, P.

Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups
Given a group automorphism $\phi:\Gamma\longrightarrow \Gamma$, one has
an action of $\Gamma$ on itself by $\phi$twisted conjugacy, namely, $g.x=gx\phi(g^{1})$.
The orbits of this action are called $\phi$twisted conjugacy classes. One says
that $\Gamma$ has the $R_\infty$property if there are infinitely many $\phi$twisted conjugacy
classes for every automorphism $\phi$ of $\Gamma$. In this paper we
show that $\operatorname{SL}(n,\mathbb{Z})$ and its
congruence subgroups have the $R_\infty$property. Further we show that
any (countable) abelian extension of $\Gamma$ has the $R_\infty$property where $\Gamma$ is a torsion
free nonelementary hyperbolic group, or $\operatorname{SL}(n,\mathbb{Z}),
\operatorname{Sp}(2n,\mathbb{Z})$ or a principal congruence
subgroup of $\operatorname{SL}(n,\mathbb{Z})$ or the fundamental group of a complete Riemannian
manifold of constant negative curvature.
Keywords:twisted conjugacy classes, hyperbolic groups, lattices in Lie groups Category:20E45 

48. CMB 2012 (vol 56 pp. 630)
 Sundar, S.

Inverse Semigroups and Sheu's Groupoid for the Odd Dimensional Quantum Spheres
In this paper, we give a different proof of the fact that the odd dimensional
quantum spheres are groupoid $C^{*}$algebras. We show that the $C^{*}$algebra
$C(S_{q}^{2\ell+1})$ is generated by an inverse semigroup $T$ of partial
isometries. We show that the groupoid $\mathcal{G}_{tight}$ associated with the
inverse semigroup $T$ by Exel is exactly the same as the groupoid
considered by Sheu.
Keywords:inverse semigroups, groupoids, odd dimensional quantum spheres Categories:46L99, 20M18 

49. CMB 2011 (vol 56 pp. 395)
 Oancea, D.

Coessential Abelianization Morphisms in the Category of Groups
An epimorphism $\phi\colon G\to H$ of groups, where $G$ has rank $n$, is called
coessential if every (ordered) generating $n$tuple of $H$ can be
lifted along $\phi$ to a generating $n$tuple for $G$. We discuss this
property in the context of the category of groups, and establish a criterion
for such a group $G$ to have the property that its abelianization
epimorphism $G\to G/[G,G]$, where $[G,G]$ is the commutator subgroup, is
coessential. We give an example of a family of 2generator groups whose
abelianization epimorphism is not coessential.
This family also provides counterexamples to the generalized AndrewsCurtis conjecture.
Keywords:coessential epimorphism, Nielsen transformations, AndrewCurtis transformations Categories:20F05, 20F99, 20J15 

50. CMB 2011 (vol 55 pp. 783)
 Motallebi, M. R.; Saiflu, H.

Products and Direct Sums in Locally Convex Cones
In this paper we define lower, upper, and symmetric completeness and
discuss closure of the sets in product and direct sums. In particular,
we introduce suitable bases for these topologies, which leads us to
investigate completeness of the direct sum and its components. Some
results obtained about $X$topologies and polars of the neighborhoods.
Keywords:product and direct sum, duality, locally convex cone Categories:20K25, 46A30, 46A20 
