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76. CMB 2010 (vol 54 pp. 39)

Chapman, S. T.; García-Sánchez, P. A.; Llena, D.; Marshall, J.
 Elements in a Numerical Semigroup with Factorizations of the Same Length Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions. Keywords:numerical monoid, numerical semigroup, non-unique factorizationCategories:20M14, 20D60, 11B75

77. CMB 2009 (vol 52 pp. 598)

Moreno, M. A.; Nicola, J.; Pardo, E.; Thomas, H.
 Numerical Semigroups That Are Not Intersections of $d$-Squashed Semigroups We say that a numerical semigroup is \emph{$d$-squashed} if it can be written in the form $$S=\frac 1 N \langle a_1,\dots,a_d \rangle \cap \mathbb{Z}$$ for $N,a_1,\dots,a_d$ positive integers with $\gcd(a_1,\dots, a_d)=1$. Rosales and Urbano have shown that a numerical semigroup is 2-squashed if and only if it is proportionally modular. Recent works by Rosales \emph{et al.} give a concrete example of a numerical semigroup that cannot be written as an intersection of $2$-squashed semigroups. We will show the existence of infinitely many numerical semigroups that cannot be written as an intersection of $2$-squashed semigroups. We also will prove the same result for $3$-squashed semigroups. We conjecture that there are numerical semigroups that cannot be written as the intersection of $d$-squashed semigroups for any fixed $d$, and we prove some partial results towards this conjecture. Keywords:numerical semigroup, squashed semigroup, proportionally modular semigroupCategories:20M14, 06F05, 46L80

78. CMB 2009 (vol 52 pp. 435)

Monson, B.; Schulte, Egon
 Modular Reduction in Abstract Polytopes The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind:\ first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in $\mathbb{Z}[\tau]$ (with $\tau$ the golden ratio), to construct new regular $4$-polytopes of hyperbolic types $\{3,5,3\}$ and $\{5,3,5\}$ with automorphism groups given by finite orthogonal groups. Keywords:abstract polytopes, regular and chiral, Coxeter groups, modular reductionCategories:51M20, 20F55

79. CMB 2009 (vol 40 pp. 330)

Kapovich, Ilya
 Amalgamated products and the Howson property We show that if $A$ is a torsion-free word hyperbolic group which belongs to class $(Q)$, that is all finitely generated subgroups of $A$ are quasiconvex in $A$, then any maximal cyclic subgroup $U$ of $A$ is a Burns subgroup of $A$. This, in particular, implies that if $B$ is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated) then $A\ast_U B$, $\langle A,t \mid U^t=V\rangle$ are also Howson groups. Finitely generated free groups, fundamental groups of closed hyperbolic surfaces and some interesting $3$-manifold groups are known to belong to class $(Q)$ and our theorem applies to them. We also describe a large class of word hyperbolic groups which are not Howson. Categories:20E06, 20E07, 20F32

80. CMB 2009 (vol 40 pp. 341)

Lee, Hyang-Sook
 The stable and unstable types of classifying spaces The main purpose of this paper is to study groups $G_1$, $G_2$ such that $H^\ast(BG_1,{\bf Z}/p)$ is isomorphic to $H^\ast(BG_2,{\bf Z}/p)$ in ${\cal U}$, the category of unstable modules over the Steenrod algebra ${\cal A}$, but not isomorphic as graded algebras over ${\bf Z}/p$. Categories:55R35, 20J06

81. CMB 2009 (vol 40 pp. 352)

Liriano, Sal
 A New Proof of a Theorem of Magnus Using naive algebraic geometric methods a new proof of the following celebrated theorem of Magnus is given: Let $G$ be a group with a presentation having $n$ generators and $m$ relations. If $G$ also has a presentation on $n-m$ generators, then $G$ is free of rank $n-m$. Categories:20E05, 20C99, 14Q99

82. CMB 2009 (vol 52 pp. 273)

MacDonald, John; Scull, Laura
 Amalgamations of Categories We consider the pushout of embedding functors in $\Cat$, the category of small categories. We show that if the embedding functors satisfy a 3-for-2 property, then the induced functors to the pushout category are also embeddings. The result follows from the connectedness of certain associated slice categories. The condition is motivated by a similar result for maps of semigroups. We show that our theorem can be applied to groupoids and to inclusions of full subcategories. We also give an example to show that the theorem does not hold when the property only holds for one of the inclusion functors, or when it is weakened to a one-sided condition. Keywords:category, pushout, amalgamationCategories:18A30, 18B40, 20L17

83. CMB 2009 (vol 52 pp. 245)

Goodaire, Edgar G.; Milies, César Polcino
 Involutions of RA Loops Let $L$ be an RA loop, that is, a loop whose loop ring over any coefficient ring $R$ is an alternative, but not associative, ring. Let $\ell\mapsto\ell^\theta$ denote an involution on $L$ and extend it linearly to the loop ring $RL$. An element $\alpha\in RL$ is \emph{symmetric} if $\alpha^\theta=\alpha$ and \emph{skew-symmetric} if $\alpha^\theta=-\alpha$. In this paper, we show that there exists an involution making the symmetric elements of $RL$ commute if and only if the characteristic of $R$ is $2$ or $\theta$ is the canonical involution on $L$, and an involution making the skew-symmetric elements of $RL$ commute if and only if the characteristic of $R$ is $2$ or $4$. Categories:20N05, 17D05

84. CMB 2009 (vol 40 pp. 47)

Hartl, Manfred
 A universal coefficient decomposition for subgroups induced by submodules of group algebras Dimension subgroups and Lie dimension subgroups are known to satisfy a universal coefficient decomposition', {\it i.e.} their value with respect to an arbitrary coefficient ring can be described in terms of their values with respect to the universal' coefficient rings given by the cyclic groups of infinite and prime power order. Here this fact is generalized to much more general types of induced subgroups, notably covering Fox subgroups and relative dimension subgroups with respect to group algebra filtrations induced by arbitrary $N$-series, as well as certain common generalisations of these which occur in the study of the former. This result relies on an extension of the principal universal coefficient decomposition theorem on polynomial ideals (due to Passi, Parmenter and Seghal), to all additive subgroups of group rings. This is possible by using homological instead of ring theoretical methods. Keywords:induced subgroups, group algebras, Fox subgroups, relative dimension, subgroups, polynomial idealsCategories:20C07, 16A27

85. CMB 2009 (vol 40 pp. 266)

Bechtell, H.; Deaconescu, M.; Silberberg, Gh.
 Finite groups with large automizers for their Abelian subgroups This note contains the classification of the finite groups $G$ satisfying the condition $N_{G}(H)/C_{G}(H)\cong \Aut(H)$ for every abelian subgroup $H$ of $G$. Categories:20E34, 20D45

86. CMB 2009 (vol 52 pp. 9)

Chassé, Dominique; Saint-Aubin, Yvan
 On the Spectrum of an $n!\times n!$ Matrix Originating from Statistical Mechanics Let $R_n(\alpha)$ be the $n!\times n!$ matrix whose matrix elements $[R_n(\alpha)]_{\sigma\rho}$, with $\sigma$ and $\rho$ in the symmetric group $\sn$, are $\alpha^{\ell(\sigma\rho^{-1})}$ with $0<\alpha<1$, where $\ell(\pi)$ denotes the number of cycles in $\pi\in \sn$. We give the spectrum of $R_n$ and show that the ratio of the largest eigenvalue $\lambda_0$ to the second largest one (in absolute value) increases as a positive power of $n$ as $n\rightarrow \infty$. Keywords:symmetric group, representation theory, eigenvalue, statistical physicsCategories:20B30, 20C30, 15A18, 82B20, 82B28

87. CMB 2008 (vol 51 pp. 584)

Purbhoo, Kevin; Willigenburg, Stephanie van
 On Tensor Products of Polynomial Representations We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of $\GL(n,\mathbb{C})$ is isomorphic to another. As a consequence we discover families of Littlewood--Richardson coefficients that are non-zero, and a condition on Schur non-negativity. Keywords:polynomial representation, symmetric function, Littlewood--Richardson coefficient, Schur non-negativeCategories:05E05, 05E10, 20C30

88. CMB 2008 (vol 51 pp. 81)

Kassel, Christian
 Homotopy Formulas for Cyclic Groups Acting on Rings The positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff. Keywords:group cohomology, norm map, cyclic group, homotopyCategories:20J06, 20K01, 16W22, 18G35

89. CMB 2008 (vol 51 pp. 134)

Rosales, J. C.; Garc\'{\i}a-Sánchez, P. A.
 Numerical Semigroups Having a Toms Decomposition We show that the class of system proportionally modular numerical semigroups coincides with the class of numerical semigroups having a Toms decomposition. Categories:20M14, 11D75

90. CMB 2008 (vol 51 pp. 114)

Petrov, V.; Semenov, N.; Zainoulline, K.
 Zero Cycles on a Twisted Cayley Plane Let $k$ be a field of characteristic not $2,3$. Let $G$ be an exceptional simple algebraic group over $k$ of type $\F$, $^1{\E_6}$ or $\E_7$ with trivial Tits algebras. Let $X$ be a projective $G$-homogeneous variety. If $G$ is of type $\E_7$, we assume in addition that the respective parabolic subgroup is of type $P_7$. The main result of the paper says that the degree map on the group of zero cycles of $X$ is injective. Categories:20G15, 14C15

91. CMB 2007 (vol 50 pp. 535)

Hohlweg, Christophe
 Generalized Descent Algebras If $A$ is a subset of the set of reflections of a finite Coxeter group $W$, we define a sub-$\ZM$-module $\DC_A(W)$ of the group algebra $\ZM W$. We discuss cases where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra, the group algebra and, if $W$ is of type $B$, the Mantaci--Reutenauer algebra. Keywords:Coxeter group, Solomon descent algebra, descent setCategories:20F55, 05E15

92. CMB 2007 (vol 50 pp. 632)

Zelenyuk, Yevhen; Zelenyuk, Yuliya
 Transformations and Colorings of Groups Let $G$ be a compact topological group and let $f\colon G\to G$ be a continuous transformation of $G$. Define $f^*\colon G\to G$ by $f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume that $H=\Imag f^*$ is a subgroup of $G$ and for every measurable $C\subseteq H$, $\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable $C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that $f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$. Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion, Categories:05D10, 20D60, 22A10

93. CMB 2007 (vol 50 pp. 206)

Golasiński, Marek; Gonçalves, Daciberg Lima
 Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$ Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times \SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional $CW$-complex of the homotopy type of an $n$-sphere. We study the automorphism group $\Aut (G)$ in order to compute the number of distinct homotopy types of spherical space forms with respect to free and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$ is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as well. Keywords:automorphism group, $CW$-complex, free and cellular $G$-action, group of self homotopy equivalences, Lyndon--Hochschild--Serre spectral sequence, special (linear) group, spherical space formCategories:55M35, 55P15, 20E22, 20F28, 57S17

94. CMB 2007 (vol 50 pp. 268)

Manuilov, V.; Thomsen, K.
 On the Lack of Inverses to $C^*$-Extensions Related to Property T Groups Using ideas of S. Wassermann on non-exact $C^*$-algebras and property T groups, we show that one of his examples of non-invertible $C^*$-extensions is not semi-invertible. To prove this, we show that a certain element vanishes in the asymptotic tensor product. We also show that a modification of the example gives a $C^*$-extension which is not even invertible up to homotopy. Keywords:$C^*$-algebra extension, property T group, asymptotic tensor $C^*$-norm, homotopyCategories:19K33, 46L06, 46L80, 20F99

95. CMB 2006 (vol 49 pp. 347)

Ecker, Jürgen
 Affine Completeness of Generalised Dihedral Groups In this paper we study affine completeness of generalised dihedral groups. We give a formula for the number of unary compatible functions on these groups, and we characterise for every $k \in~\N$ the $k$-affine complete generalised dihedral groups. We find that the direct product of a $1$-affine complete group with itself need not be $1$-affine complete. Finally, we give an example of a nonabelian solvable affine complete group. For nilpotent groups we find a strong necessary condition for $2$-affine completeness. Categories:08A40, 16Y30, 20F05

96. CMB 2006 (vol 49 pp. 196)

 Another Proof of Totaro's Theorem on $E_8$-Torsors We give a short proof of Totaro's theorem that every$E_8$-torsor over a field $k$ becomes trivial over a finiteseparable extension of $k$of degree dividing $d(E_8)=2^63^25$. Categories:11E72, 14M17, 20G15

97. CMB 2006 (vol 49 pp. 296)

Sch"utt, Matthias
 On the Modularity of Three Calabi--Yau Threefolds With Bad Reduction at 11 This paper investigates the modularity of three non-rigid Calabi--Yau threefolds with bad reduction at 11. They are constructed as fibre products of rational elliptic surfaces, involving the modular elliptic surface of level 5. Their middle $\ell$-adic cohomology groups are shown to split into two-dimensional pieces, all but one of which can be interpreted in terms of elliptic curves. The remaining pieces are associated to newforms of weight 4 and level 22 or 55, respectively. For this purpose, we develop a method by Serre to compare the corresponding two-dimensional 2-adic Galois representations with uneven trace. Eventually this method is also applied to a self fibre product of the Hesse-pencil, relating it to a newform of weight 4 and level 27. Categories:14J32, 11F11, 11F23, 20C12

98. CMB 2006 (vol 49 pp. 285)

Riedl, Jeffrey M.
 Orbits and Stabilizers for Solvable Linear Groups We extend a result of Noritzsch, which describes the orbit sizes in the action of a Frobenius group $G$ on a finite vector space $V$ under certain conditions, to a more general class of finite solvable groups $G$. This result has applications in computing irreducible character degrees of finite groups. Another application, proved here, is a result concerning the structure of certain groups with few complex irreducible character degrees. Categories:20B99, 20C15, 20C20

99. 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

100. CMB 2006 (vol 49 pp. 127)

Lewis, Mark L.
 Character Degree Graphs of Solvable Groups of Fitting Height $2$ Given a finite group $G$, we attach to the character degrees of $G$ a graph whose vertex set is the set of primes dividing the degrees of irreducible characters of $G$, and with an edge between $p$ and $q$ if $pq$ divides the degree of some irreducible character of $G$. In this paper, we describe which graphs occur when $G$ is a solvable group of Fitting height $2$. Category:20C15
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