Expand all Collapse all | Results 1 - 25 of 109 |
1. CMB Online first
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 |
2. CMB Online first
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 |
3. CMB Online first
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 |
4. CMB 2014 (vol 58 pp. 105)
On Graphs Associated with Character Degrees and Conjugacy Class Sizes of Direct Products of Finite Groups |
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 |
5. CMB 2014 (vol 58 pp. 182)
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 |
6. CMB 2014 (vol 58 pp. 196)
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 domain Categories:20H25, 57M60 |
7. CMB Online first
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 |
8. CMB 2014 (vol 57 pp. 884)
$m$-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 subgroup Categories:20D10, 20D15 |
9. CMB 2014 (vol 57 pp. 621)
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 |
10. CMB 2014 (vol 57 pp. 708)
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 |
11. CMB 2014 (vol 57 pp. 648)
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 |
12. CMB 2014 (vol 57 pp. 277)
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 |
13. CMB 2014 (vol 57 pp. 390)
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, building Categories:20G44, 20E42, 51E24 |
14. CMB 2014 (vol 57 pp. 231)
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 |
15. CMB 2013 (vol 57 pp. 506)
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 |
16. CMB 2013 (vol 57 pp. 449)
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 |
17. CMB 2013 (vol 57 pp. 125)
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 |
18. CMB 2013 (vol 57 pp. 9)
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 algebra Category:20C99 |
19. CMB 2013 (vol 56 pp. 795)
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 position Categories:20G15, 20G41 |
20. CMB 2012 (vol 57 pp. 97)
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, rationality Categories:20G15, 14L24 |
21. CMB 2012 (vol 57 pp. 303)
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, descent Categories:14L24, 20G41 |
22. CMB 2012 (vol 56 pp. 881)
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 group Categories:30F40, 22E40, 20H10 |
23. CMB 2012 (vol 57 pp. 326)
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-divisors Categories:20C07, 20E06, 20F05, , 20F50 |
24. CMB 2012 (vol 57 pp. 424)
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 |
25. CMB 2012 (vol 57 pp. 132)
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 groups Category:20E45 |