Search results for "Combinatorics"
showing 10 items of 1770 documents
New Refinements of the McKay Conjecture for Arbitrary Finite Groups
2004
Let $G$ be an arbitrary finite group and fix a prime number $p$. The McKay conjecture asserts that $G$ and the normalizer in $G$ of a Sylow $p$-subgroup have equal numbers of irreducible characters with degrees not divisible by $p$. The Alperin-McKay conjecture is a version of this as applied to individual Brauer $p$-blocks of $G$. We offer evidence that perhaps much stronger forms of both of these conjectures are true.
Conjugacy classes, characters and products of elements
2019
Recently, Baumslag and Wiegold proved that a finite group $G$ is nilpotent if and only if $o(xy)=o(x)o(y)$ for every $x,y\in G$ of coprime order. Motivated by this result, we study the groups with the property that $(xy)^G=x^Gy^G$ and those with the property that $\chi(xy)=\chi(x)\chi(y)$ for every complex irreducible character $\chi$ of $G$ and every nontrivial $x, y \in G$ of pairwise coprime order. We also consider several ways of weakening the hypothesis on $x$ and $y$. While the result of Baumslag and Wiegold is completely elementary, some of our arguments here depend on (parts of) the classification of finite simple groups.
Almost polynomial growth: Classifying varieties of graded algebras
2015
Let G be a finite group, V a variety of associative G-graded algebras and c (V), n = 1, 2, …, its sequence of graded codimensions. It was recently shown by Valenti that such a sequence is polynomially bounded if and only if V does not contain a finite list of G-graded algebras. The list consists of group algebras of groups of order a prime number, the infinite-dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with suitable gradings. Such algebras generate the only varieties of G-graded algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety is polynomially bounded. In this paper we completely classify all sub…
Degrees of rational characters of finite groups
2010
Abstract A classical theorem of John Thompson on character degrees states that if the degree of any complex irreducible character of a finite group G is 1 or divisible by a prime p, then G has a normal p-complement. In this paper, we consider fields of values of characters and prove some improvements of this result.
Finite groups with real-valued irreducible characters of prime degree
2008
Abstract In this paper we describe the structure of finite groups whose real-valued nonlinear irreducible characters have all prime degree. The more general situation in which the real-valued irreducible characters of a finite group have all squarefree degree is also considered.
On finite soluble groups in which Sylow permutability is a transitive relation
2003
A characterisation of finite soluble groups in which Sylow permutability is a transitive relation by means of subgroup embedding properties enjoyed by all the subgroups is proved in the paper. The key point is an extension of a subnormality criterion due to Wielandt.
Poincaré inequalities and Steiner symmetrization
1996
A complete geometric characterization for a general Steiner symmetric domain Ω ⊂ Rn to satisfy the Poincare inequality with exponent p > n−1 is obtained and it is shown that this range of exponents is best possible. In the case where the Steiner symmetric domain is determined by revolving the graph of a Lipschitz continuous function, it is shown that the preceding characterization works for all p > 1 and furthermore for such domains a geometric characterization for a more general Sobolev–Poincare inequality to hold is given. Although the operation of Steiner symmetrization need not always preserve a Poincare inequality, a general class of domains is given for which Poincare inequalities are…
Critical points of higher order for the normal map of immersions in Rd
2012
We study the critical points of the normal map v : NM -> Rk+n, where M is an immersed k-dimensional submanifold of Rk+n, NM is the normal bundle of M and v(m, u) = m + u if u is an element of NmM. Usually, the image of these critical points is called the focal set. However, in that set there is a subset where the focusing is highest, as happens in the case of curves in R-3 with the curve of the centers of spheres with contact of third order with the curve. We give a definition of r-critical points of a smooth map between manifolds, and apply it to study the 2 and 3-critical points of the normal map in general and the 2-critical points for the case k = n = 2 in detail. In the later case we a…
Lie algebra on the transverse bundle of a decreasing family of foliations
2010
Abstract J. Lehmann-Lejeune in [J. Lehmann-Lejeune, Cohomologies sur le fibre transverse a un feuilletage, C.R.A.S. Paris 295 (1982), 495–498] defined on the transverse bundle V to a foliation on a manifold M, a zero-deformable structure J such that J 2 = 0 and for every pair of vector fields X , Y on M: [ J X , J Y ] − J [ J X , Y ] − J [ X , J Y ] + J 2 [ X , Y ] = 0 . For every open set Ω of V, J. Lehmann-Lejeune studied the Lie Algebra L J ( Ω ) of vector fields X defined on Ω such that the Lie derivative L ( X ) J is equal to zero i.e., for each vector field Y on Ω : [ X , J Y ] = J [ X , Y ] and showed that for every vector field X on Ω such that X ∈ K e r J , we can write X = ∑ [ Y ,…
The Intersection of $3$-Maximal Submonids
2020
Very little is known about the structure of the intersection of two $k$-generated monoids of words, even for $k=3$. Here we investigate the case of $k$-maximal monoids, that is, monoids whose basis of cardinality $k$ cannot be non-trivially decomposed into at most $k$ words. We characterize the intersection in the case of two $3$-maximal monoids.