Search results for "combinatoric"
showing 10 items of 1776 documents
Quadratic characters in groups of odd order
2009
Abstract We prove that in a finite group of odd order, the number of irreducible quadratic characters is the number of quadratic conjugacy classes.
On sigma-subnormal subgroups of factorised finite groups
2020
Abstract Let σ = { σ i : i ∈ I } be a partition of the set P of all prime numbers. A subgroup X of a finite group G is called σ-subnormal in G if there is chain of subgroups X = X 0 ⊆ X 1 ⊆ ⋯ ⊆ X n = G with X i − 1 normal in X i or X i / C o r e X i ( X i − 1 ) is a σ i -group for some i ∈ I , 1 ≤ i ≤ n . In the special case that σ is the partition of P into sets containing exactly one prime each, the σ-subnormality reduces to the familiar case of subnormality. If a finite soluble group G = A B is factorised as the product of the subgroups A and B, and X is a subgroup of G such that X is σ-subnormal in 〈 X , X g 〉 for all g ∈ A ∪ B , we prove that X is σ-subnormal in G. This is an extension…
A characterisation of nilpotent blocks
2015
Let $B$ be a $p$-block of a finite group, and set $m=$ $\sum \chi(1)^2$, the sum taken over all height zero characters of $B$. Motivated by a result of M. Isaacs characterising $p$-nilpotent finite groups in terms of character degrees, we show that $B$ is nilpotent if and only if the exact power of $p$ dividing $m$ is equal to the $p$-part of $|G:P|^2|P:R|$, where $P$ is a defect group of $B$ and where $R$ is the focal subgroup of $P$ with respect to a fusion system $\CF$ of $B$ on $P$. The proof involves the hyperfocal subalgebra $D$ of a source algebra of $B$. We conjecture that all ordinary irreducible characters of $D$ have degree prime to $p$ if and only if the $\CF$-hyperfocal subgrou…
Non-vanishing elements of finite groups
2010
AbstractLet G be a finite group, and let Irr(G) denote the set of irreducible complex characters of G. An element x of G is non-vanishing if, for every χ in Irr(G), we have χ(x)≠0. We prove that, if x is a non-vanishing element of G and the order of x is coprime to 6, then x lies in the Fitting subgroup of G.
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.