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1. CMB Online first

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 $RA$ loopsCategories:20N05, 17D05

2. CMB Online first

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, invariantsCategories:16T20, 17B37, 20G42

3. CMB Online first

Chen, Guiyun; Li, Lili
 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 groupCategory:20D15

4. CMB Online first

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

5. CMB 2014 (vol 58 pp. 105)

 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 operationCategories:20E45, 05C25, 05C76

6. 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, crownsCategories:20D30, 20D60, 20E15

7. CMB 2014 (vol 58 pp. 196)

Yang, Qingjie; Zhong, Weiting
 Dihedral Groups of order $2p$ of Automorphisms of Compact Riemann Surfaces of Genus $p-1$ In this paper we prove that there is only one conjugacy class of dihedral group of order $2p$ in the $2(p-1)\times 2(p-1)$ integral symplectic group can be realized by an analytic automorphism group of compact connected Riemann surfaces of genus $p-1$. A pair of representative generators of the realizable class is also given. Keywords:dihedral group, automorphism group, Riemann surface, integral symplectic matrix, fundamental domainCategories:20H25, 57M60

8. 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 $G_{i-1}\leq K \lt H\leq G_i$, 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 subgroupCategories:20D10, 20D15

9. 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 semigroupCategory:20M18

10. 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 decayCategories:46L54, 20G42, 46L65

11. 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, formationCategories:20D10, 20D20

12. 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 latticesCategories:20D10, 20D20, 20E15, 20F16

13. 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 Kac-Moody 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:Kac-Moody group, twin tree, simplicity, root system, buildingCategories:20G44, 20E42, 51E24

14. 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 characterCategories:20C99, 16G30

15. 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 categoriesCategories:20F36, 16W30, 18D10

16. 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 groupsCategories:43A20, 20C15

17. 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 groupsCategory:20D15

18. CMB 2013 (vol 57 pp. 9)

Alperin, Roger C.; Peterson, Brian L.
 Integral Sets and the Center of a Finite Group We give a description of the atoms in the Boolean algebra generated by the integral subsets of a finite group. Keywords:integral set, characters, Boolean algebraCategory:20C99

19. 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 2-dimension 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 positionCategories:20G15, 20G41

20. CMB 2012 (vol 57 pp. 97)

Levy, Jason
 Rationality and the Jordan-Gatti-Viniberghi decomposition We verify our earlier conjecture and use it to prove that the semisimple parts of the rational Jordan-Kac-Vinberg decompositions of a rational vector all lie in a single rational orbit. Keywords:reductive group, $G$-module, Jordan decomposition, orbit closure, rationalityCategories:20G15, 14L24

21. CMB 2012 (vol 57 pp. 303)

Gille, Philippe
 Octonion Algebras over Rings are not Determined by their Norms Answering a question of H. Petersson, we provide a class of examples of pair of octonion algebras over a ring having isometric norms. Keywords:octonion algebras, torsors, descentCategories:14L24, 20G41

22. CMB 2012 (vol 56 pp. 881)

Xie, BaoHua; Wang, JieYan; Jiang, YuePing
 Free Groups Generated by Two Heisenberg Translations In this paper, we will discuss the groups generated by two Heisenberg translations of $\mathbf{PU}(2,1)$ and determine when they are free. Keywords:free group, Heisenberg group, complex triangle groupCategories:30F40, 22E40, 20H10

23. CMB 2012 (vol 57 pp. 326)

Ivanov, S. V.; Mikhailov, Roman
 On Zero-divisors in Group Rings of Groups with Torsion Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n \gg 1$ is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion. Keywords:Burnside groups, free products of groups, group rings, zero-divisorsCategories:20C07, 20E06, 20F05, , 20F50

24. 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 injectivityCategories:20G42, 22D25, 46L89

25. 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 non-elementary 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 groupsCategory:20E45
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