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76. CMB 2006 (vol 49 pp. 96)

Külshammer, Burkhard
Roots of Simple Modules
We introduce roots of indecomposable modules over group algebras of finite groups, and we investigate some of their properties. This allows us to correct an error in Landrock's book which has to do with roots of simple modules.

Categories:20C20, 20C05

77. CMB 2005 (vol 48 pp. 460)

Sommers, Eric N.
$B$-Stable Ideals in the Nilradical of a Borel Subalgebra
We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also count the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.

Categories:20F55, 17B20, 05E99

78. CMB 2005 (vol 48 pp. 211)

Germain, Jam
The Distribution of Totatives
The integers coprime to $n$ are called the {\it totatives} \rm of $n$. D. H. Lehmer and Paul Erd\H{o}s were interested in understanding when the number of totatives between $in/k$ and $(i+1)n/k$ are $1/k$th of the total number of totatives up to $n$. They provided criteria in various cases. Here we give an ``if and only if'' criterion which allows us to recover most of the previous results in this literature and to go beyond, as well to reformulate the problem in terms of combinatorial group theory. Our criterion is that the above holds if and only if for every odd character $\chi \pmod \kappa$ (where $\kappa:=k/\gcd(k,n/\prod_{p|n} p)$) there exists a prime $p=p_\chi$ dividing $n$ for which $\chi(p)=1$.

Categories:11A05, 11A07, 11A25, 20C99

79. CMB 2005 (vol 48 pp. 32)

Dąbkowski, Mieczysław K.; Przytycki, Józef H.; Togha, Amir A.
Non-Left-Orderable 3-Manifold Groups
We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of $S^3$ branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of $S^3$ branched along various hyperbolic 2-bridge knots. %with various hyperbolic 2-bridge knots as branched sets. The manifold obtained in such a way from the $5_2$ knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

Categories:57M25, 57M12, 20F60

80. CMB 2005 (vol 48 pp. 80)

Herman, Allen; Li, Yuanlin; Parmenter, M. M.
Trivial Units for Group Rings with $G$-adapted Coefficient Rings
For each finite group $G$ for which the integral group ring $\mathbb{Z}G$ has only trivial units, we give ring-theoretic conditions for a commutative ring $R$ under which the group ring $RG$ has nontrivial units. Several examples of rings satisfying the conditions and rings not satisfying the conditions are given. In addition, we extend a well-known result for fields by showing that if $R$ is a ring of finite characteristic and $RG$ has only trivial units, then $G$ has order at most 3.

Categories:16S34, 16U60, 20C05

81. CMB 2005 (vol 48 pp. 41)

Dixon, John D.; Barghi, A. Rahnamai
Degree Homogeneous Subgroups
Let $G$ be a finite group and $H$ be a subgroup. We say that $H$ is \emph{degree homogeneous }if, for each $\chi\in \Irr(G)$, all the irreducible constituents of the restriction $\chi_{H}$ have the same degree. Subgroups which are either normal or abelian are obvious examples of degree homogeneous subgroups. Following a question by E.~M. Zhmud', we investigate general properties of such subgroups. It appears unlikely that degree homogeneous subgroups can be characterized entirely by abstract group properties, but we provide mixed criteria (involving both group structure and character properties) which are both necessary and sufficient. For example, $H$ is degree homogeneous in $G$ if and only if the derived subgroup $H^{\prime}$ is normal in $G$ and, for every pair $\alpha,\beta$ of irreducible $G$-conjugate characters of $H^{\prime}$, all irreducible constituents of $\alpha^{H}$ and $\beta^{H}$ have the same degree.

Category:20C15

82. CMB 2004 (vol 47 pp. 530)

Iranmanesh, A.; Khosravi, B.
A Characterization of $ PSU_{11}(q)$
Order components of a finite simple group were introduced in [4]. It was proved that some non-abelian simple groups are uniquely determined by their order components. As the main result of this paper, we show that groups $PSU_{11}(q)$ are also uniquely determined by their order components. As corollaries of this result, the validity of a conjecture of J. G. Thompson and a conjecture of W. Shi and J. Bi both on $PSU_{11}(q)$ are obtained.

Keywords:Prime graph, order component, finite group,simple group
Categories:20D08, 20D05, 20D60

83. CMB 2004 (vol 47 pp. 439)

Parker, John R.
On the Stable Basin Theorem
The stable basin theorem was introduced by Basmajian and Miner as a key step in their necessary condition for the discreteness of a non-elementary group of complex hyperbolic isometries. In this paper we improve several of Basmajian and Miner's key estimates and so give a substantial improvement on the main inequality in the stable basin theorem.

Categories:22E40, 20H10, 57S30

84. CMB 2004 (vol 47 pp. 343)

Drensky, Vesselin; Hammoudi, Lakhdar
Combinatorics of Words and Semigroup Algebras Which Are Sums of Locally Nilpotent Subalgebras
We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. Like all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words.

Keywords:locally nilpotent rings,, nil rings, locally nilpotent semigroups,, semigroup algebras, monomial algebras, infinite words
Categories:16N40, 16S15, 20M05, 20M25, 68R15

85. CMB 2004 (vol 47 pp. 237)

Laubie, François
Ramification des séries formelles
Let $p$ be a prime number. Let $k$ be a finite field of characteristic $p$. The subset $X+X^2 k[[X]]$ of the ring $k[[X]]$ is a group under the substitution law $\circ $ sometimes called the Nottingham group of $k$; it is denoted by $\mathcal{R}_k$. The ramification of one series $\gamma\in\mathcal{R}_k$ is caracterized by its lower ramification numbers: $i_m(\gamma)=\ord_X \bigl(\gamma^{p^m} (X)/X - 1\bigr)$, as well as its upper ramification numbers: $$ u_m (\gamma) = i_0 (\gamma) + \frac{i_1 (\gamma) - i_0(\gamma)}{p} + \cdots + \frac{i_m (\gamma) - i_{m-1} (\gamma)}{p^m} , \quad (m \in \mathbb{N}). $$ By Sen's theorem, the $u_m(\gamma)$ are integers. In this paper, we determine the sequences of integers $(u_m)$ for which there exists $\gamma\in\mathcal{R}_k$ such that $u_m(\gamma)=u_m$ for all integer $m \geq 0$.

Keywords:ramification, Nottingham group
Categories:11S15, 20E18

86. CMB 2004 (vol 47 pp. 298)

Yahaghi, Bamdad R.
Near Triangularizability Implies Triangularizability
In this paper we consider collections of compact operators on a real or complex Banach space including linear operators on finite-dimensional vector spaces. We show that such a collection is simultaneously triangularizable if and only if it is arbitrarily close to a simultaneously triangularizable collection of compact operators. As an application of these results we obtain an invariant subspace theorem for certain bounded operators. We further prove that in finite dimensions near reducibility implies reducibility whenever the ground field is $\BR$ or $\BC$.

Keywords:Linear transformation, Compact operator,, Triangularizability, Banach space, Hilbert, space
Categories:47A15, 47D03, 20M20

87. CMB 2004 (vol 47 pp. 161)

Alspach, Brian; Du, Shaofei
Suborbit Structure of Permutation $p$-Groups and an Application to Cayley Digraph Isomorphism
Let $P$ be a transitive permutation group of order $p^m$, $p$ an odd prime, containing a regular cyclic subgroup. The main result of this paper is a determination of the suborbits of $P$. The main result is used to give a simple proof of a recent result by J.~Morris on Cayley digraph isomorphisms.

Categories:20B25, 05C60

88. CMB 2003 (vol 46 pp. 509)

Benson, David J.; Kumjian, Alex; Phillips, N. Christopher
Symmetries of Kirchberg Algebras
Let $G_0$ and $G_1$ be countable abelian groups. Let $\gamma_i$ be an automorphism of $G_i$ of order two. Then there exists a unital Kirchberg algebra $A$ satisfying the Universal Coefficient Theorem and with $[1_A] = 0$ in $K_0 (A)$, and an automorphism $\alpha \in \Aut(A)$ of order two, such that $K_0 (A) \cong G_0$, such that $K_1 (A) \cong G_1$, and such that $\alpha_* \colon K_i (A) \to K_i (A)$ is $\gamma_i$. As a consequence, we prove that every $\mathbb{Z}_2$-graded countable module over the representation ring $R (\mathbb{Z}_2)$ of $\mathbb{Z}_2$ is isomorphic to the equivariant $K$-theory $K^{\mathbb{Z}_2} (A)$ for some action of $\mathbb{Z}_2$ on a unital Kirchberg algebra~$A$. Along the way, we prove that every not necessarily finitely generated $\mathbb{Z} [\mathbb{Z}_2]$-module which is free as a $\mathbb{Z}$-module has a direct sum decomposition with only three kinds of summands, namely $\mathbb{Z} [\mathbb{Z}_2]$ itself and $\mathbb{Z}$ on which the nontrivial element of $\mathbb{Z}_2$ acts either trivially or by multiplication by $-1$.

Categories:20C10, 46L55, 19K99, 19L47, 46L40, 46L80

89. CMB 2003 (vol 46 pp. 332)

Đoković, Dragomir Z.; Tam, Tin-Yau
Some Questions about Semisimple Lie Groups Originating in Matrix Theory
We generalize the well-known result that a square traceless complex matrix is unitarily similar to a matrix with zero diagonal to arbitrary connected semisimple complex Lie groups $G$ and their Lie algebras $\mathfrak{g}$ under the action of a maximal compact subgroup $K$ of $G$. We also introduce a natural partial order on $\mathfrak{g}$: $x\le y$ if $f(K\cdot x) \subseteq f(K\cdot y)$ for all $f\in \mathfrak{g}^*$, the complex dual of $\mathfrak{g}$. This partial order is $K$-invariant and induces a partial order on the orbit space $\mathfrak{g}/K$. We prove that, under some restrictions on $\mathfrak{g}$, the set $f(K\cdot x)$ is star-shaped with respect to the origin.

Categories:15A45, 20G20, 22E60

90. CMB 2003 (vol 46 pp. 310)

Wang, Xiaofeng
Second Order Dehn Functions of Asynchronously Automatic Groups
Upper bounds of second order Dehn functions of asynchronously automatic groups are obtained.

Keywords:second order Dehn function, combing, asynchronously automatic group
Categories:20E06, 20F05, 57M05

91. CMB 2003 (vol 46 pp. 299)

Tomaszewski, Witold
A Basis of Bachmuth Type in the Commutator Subgroup of a Free Group
We show here that the commutator subgroup of a free group of finite rank poses a basis of Bachmuth's type.

Categories:20E05, 20F12, 20F05

92. CMB 2003 (vol 46 pp. 268)

Puls, Michael J.
Group Cohomology and $L^p$-Cohomology of Finitely Generated Groups
Let $G$ be a finitely generated, infinite group, let $p>1$, and let $L^p(G)$ denote the Banach space $\{ \sum_{x\in G} a_xx \mid \sum_{x\in G} |a_x |^p < \infty \}$. In this paper we will study the first cohomology group of $G$ with coefficients in $L^p(G)$, and the first reduced $L^p$-cohomology space of $G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.

Keywords:group cohomology, $L^p$-cohomology, central element of infinite order, harmonic function, continuous linear functional
Categories:43A15, 20F65, 20F18

93. CMB 2003 (vol 46 pp. 204)

Levy, Jason
Rationality and Orbit Closures
Suppose we are given a finite-dimensional vector space $V$ equipped with an $F$-rational action of a linearly algebraic group $G$, with $F$ a characteristic zero field. We conjecture the following: to each vector $v\in V(F)$ there corresponds a canonical $G(F)$-orbit of semisimple vectors of $V$. In the case of the adjoint action, this orbit is the $G(F)$-orbit of the semisimple part of $v$, so this conjecture can be considered a generalization of the Jordan decomposition. We prove some cases of the conjecture.

Categories:14L24, 20G15

94. CMB 2003 (vol 46 pp. 140)

Renner, Lex E.
An Explicit Cell Decomposition of the Wonderful Compactification of a Semisimple Algebraic Group
We determine an explicit cell decomposition of the wonderful compactification of a semi\-simple algebraic group. To do this we first identify the $B\times B$-orbits using the generalized Bruhat decomposition of a reductive monoid. From there we show how each cell is made up from $B\times B$-orbits.

Categories:14L30, 14M17, 20M17

95. CMB 2003 (vol 46 pp. 122)

Moon, Myoungho
On Certain Finitely Generated Subgroups of Groups Which Split
Define a group $G$ to be in the class $\mathcal{S}$ if for any finitely generated subgroup $K$ of $G$ having the property that there is a positive integer $n$ such that $g^n \in K$ for all $g\in G$, $K$ has finite index in $G$. We show that a free product with amalgamation $A*_C B$ and an $\HNN$ group $A *_C$ belong to $\mathcal{S}$, if $C$ is in $\mathcal{S}$ and every subgroup of $C$ is finitely generated.

Keywords:free product with amalgamation, $\HNN$ group, graph of groups, fundamental group
Categories:20E06, 20E08, 57M07

96. CMB 2002 (vol 45 pp. 686)

Rauschning, Jan; Slodowy, Peter
An Aspect of Icosahedral Symmetry
We embed the moduli space $Q$ of 5 points on the projective line $S_5$-equivariantly into $\mathbb{P} (V)$, where $V$ is the 6-dimensional irreducible module of the symmetric group $S_5$. This module splits with respect to the icosahedral group $A_5$ into the two standard 3-dimensional representations. The resulting linear projections of $Q$ relate the action of $A_5$ on $Q$ to those on the regular icosahedron.

Categories:14L24, 20B25

97. CMB 2002 (vol 45 pp. 537)

Chapoton, Frédéric; Fomin, Sergey; Zelevinsky, Andrei
Polytopal Realizations of Generalized Associahedra
No abstract.

Categories:05E15, 20F55, 52C07

98. CMB 2002 (vol 45 pp. 388)

Gille, Philippe
Algèbres simples centrales de degré 5 et $E_8$
As a consequence of a theorem of Rost-Springer, we establish that the cyclicity problem for central simple algebra of degree~5 on fields containg a fifth root of unity is equivalent to the study of anisotropic elements of order 5 in the split group of type~$E_8$.

Keywords:algèbres simples centrales, cohomologie galoisienne
Categories:16S35, 12G05, 20G15

99. CMB 2002 (vol 45 pp. 168)

Byott, Nigel P.; Elder, G. Griffith
Biquadratic Extensions with One Break
We explicitly describe, in terms of indecomposable $\mathbb{Z}_2 [G]$-modules, the Galois module structure of ideals in totally ramified biquadratic extensions of local number fields with only one break in their ramification filtration. This paper completes work begun in [Elder: Canad. J.~Math. (5) {\bf 50}(1998), 1007--1047].

Categories:11S15, 20C11

100. CMB 2002 (vol 45 pp. 294)

Sebbar, Abdellah
Modular Subgroups, Forms, Curves and Surfaces
We study a class of subgroups of $\PSL_2 (\mathbb{Z})$ which can be characterized in different ways, such as congruence groups, modular forms, modular curves, elliptic surfaces, lattices and graphs.

Category:20H05
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