Search results for "Brauer"
showing 7 items of 27 documents
A characteristic subgroup and kernels of Brauer characters
2005
If G is finite group and P is a Sylow p-subgroup of G, we prove that there is a unique largest normal subgroup L of G such that L ⋂ P = L ⋂ NG (P). If G is p-solvable, then L is the intersection of the kernels of the irreducible Brauer characters of G of degree not divisible by p.
Characters of relative p'-degree over normal subgroups
2013
Let Z be a normal subgroup of a finite group G , let ??Irr(Z) be an irreducible complex character of Z , and let p be a prime number. If p does not divide the integers ?(1)/?(1) for all ??Irr(G) lying over ? , then we prove that the Sylow p -subgroups of G/Z are abelian. This theorem, which generalizes the Gluck-Wolf Theorem to arbitrary finite groups, is one of the principal obstacles to proving the celebrated Brauer Height Zero Conjecture
On fully ramified Brauer characters
2014
Let Z be a normal subgroup of a finite group, let p≠5 be a prime and let λ∈IBr(Z) be an irreducible G-invariant p-Brauer character of Z. Suppose that λG=eφ for some φ∈IBr(G). Then G/Z is solvable. In other words, a twisted group algebra over an algebraically closed field of characteristic not 5 with a unique class of simple modules comes from a solvable group.
NONVANISHING ELEMENTS FOR BRAUER CHARACTERS
2015
Let $G$ be a finite group and $p$ a prime. We say that a $p$-regular element $g$ of $G$ is $p$-nonvanishing if no irreducible $p$-Brauer character of $G$ takes the value $0$ on $g$. The main result of this paper shows that if $G$ is solvable and $g\in G$ is a $p$-regular element which is $p$-nonvanishing, then $g$ lies in a normal subgroup of $G$ whose $p$-length and $p^{\prime }$-length are both at most 2 (with possible exceptions for $p\leq 7$), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for $p>7$) some results in Dolfi and Pacifici [‘Zero…
Field of values and Brauer characters of q′-degree
2007
Abstract We study the existence of non-trivial real-valued (respectively rational-valued) irreducible p -Brauer characters of degree not divisible by q .
Brauer characters with cyclotomic field of values
2008
It has been shown in an earlier paper [G. Navarro, Pham Huu Tiep, Rational Brauer characters, Math. Ann. 335 (2006) 675–686] that, for any odd prime p, every finite group of even order has a non-trivial rational-valued irreducible p-Brauer character. For p=2 this statement is no longer true. In this paper we determine the possible non-abelian composition factors of finite groups without non-trivial rational-valued irreducible 2-Brauer characters. We also prove that, if p≠q are primes, then any finite group of order divisible by q has a non-trivial irreducible p-Brauer character with values in the cyclotomic field Q(exp(2πi/q)).
Sylow Normalizers and Brauer Character Degrees
2000
Suppose that G is a finite group. In this note, we show that a local condition about Sylow normalizers is equivalent to a global condition on the degrees of certain irreducible Brauer characters of G. Theorem A. Let G be a finite p; q-solvable group, and let Q ∈ SylqG and P ∈ SylpG. Then every irreducible p-Brauer character of G of q′degree has p′-degree if and only if NGQ is contained in some G-conjugate of NGP. Theorem A needs a solvability hypothesis. If p = 7, then the irreducible p-Brauer characters of the group G = PSL2; 27 have degrees 1; 13; 26; 28. If we set q = 2, then each q′-degree is also a p′-degree.