Search results for "Finite group"
showing 10 items of 205 documents
The prime graph on class sizes of a finite group has a bipartite complement
2020
Abstract Let G be a finite group, and let cs ( G ) denote the set of sizes of the conjugacy classes of G. The prime graph built on cs ( G ) , that we denote by Δ ( G ) , is the (simple undirected) graph whose vertices are the prime divisors of the numbers in cs ( G ) , and two distinct vertices p, q are adjacent if and only if pq divides some number in cs ( G ) . A rephrasing of the main theorem in [8] is that the complement Δ ‾ ( G ) of the graph Δ ( G ) does not contain any cycle of length 3. In this paper we generalize this result, showing that Δ ‾ ( G ) does not contain any cycle of odd length, i.e., it is a bipartite graph. In other words, the vertex set V ( G ) of Δ ( G ) is covered b…
Groups with few $p'$-character degrees
2019
Abstract We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group G contains only two distinct values not divisible by a given prime number p > 3 , then O p p ′ p p ′ ( G ) = 1 . This is done by using the classification of finite simple groups.
Bounding the number of vertices in the degree graph of a finite group
2020
Abstract Let G be a finite group, and let cd ( G ) denote the set of degrees of the irreducible complex characters of G . The degree graph Δ ( G ) of G is defined as the simple undirected graph whose vertex set V ( G ) consists of the prime divisors of the numbers in cd ( G ) , two distinct vertices p and q being adjacent if and only if pq divides some number in cd ( G ) . In this note, we provide an upper bound on the size of V ( G ) in terms of the clique number ω ( G ) (i.e., the maximum size of a subset of V ( G ) inducing a complete subgraph) of Δ ( G ) . Namely, we show that | V ( G ) | ≤ max { 2 ω ( G ) + 1 , 3 ω ( G ) − 4 } . Examples are given in order to show that the bound is bes…
Characterizing normal Sylow p-subgroups by character degrees
2012
Abstract Suppose that G is a finite group, let p be a prime and let P ∈ Syl p ( G ) . We prove that P is normal in G if and only if all the irreducible constituents of the permutation character ( 1 P ) G have degree not divisible by p.
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.