scholarly journals HALL SUBGROUPS AND STABLE BRAUER CHARACTERS

2001 ◽  
Vol 44 (1) ◽  
pp. 111-115
Author(s):  
Gabriel Navarro
Keyword(s):  
Subject Classification ◽  
Primary 20C15 ◽  
Separable Group ◽  
Brauer Character ◽  
Hall Subgroups ◽  
Brauer Characters

AbstractLet $H$ be a Hall $\pi$-subgroup of a finite $\pi$-separable group $G$, and let $\alpha$ be an irreducible Brauer character of $H$. If $\alpha(x)=\alpha(y)$ whenever $x,y \in H$ are $p$-regular and $G$-conjugate, then $\alpha$ extends to a Brauer character of $G$.AMS 2000 Mathematics subject classification: Primary 20C15; 20C20

A Note on Brauer Character Degrees of Solvable Groups

1991 ◽  
Vol 34 (3) ◽  
pp. 423-425 ◽  
Author(s):  
You-Qiang Wang
Keyword(s):  
Solvable Group ◽  
Solvable Groups ◽  
Character Degrees ◽  
Finite Solvable Group ◽  
Prime Integer ◽  
Derived Length ◽  
Brauer Character ◽  
Brauer Characters

AbstractLet G be a finite solvable group. Fix a prime integer p and let t be the number of distinct degrees of irreducible Brauer characters of G with respect to the prime p. We obtain the bound 3t — 2 for the derived length of a Hall p'-subgroup of G. Furthermore, if |G| is odd, then the derived length of a Hall p'-subgroup of G is bounded by /.

Induction and Restriction of π-Partial Characters and their Lifts

1996 ◽  
Vol 48 (6) ◽  
pp. 1210-1223 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Subnormal Subgroup ◽  
Primitive Character ◽  
Separable Group ◽  
Ordinary Character ◽  
Brauer Characters ◽  
Alternative Set

AbstractLet G be a finite π-separable group, where π is a set of primes. The π-partial characters of G are the restrictions of the ordinary characters to the set of π-elements of G. Such an object is said to be irreducible if it is not the sum of two nonzero partial characters and the set of irreducible π- partial characters of G is denoted Iπ(G). (If p is a prime and π = p′, then Iπ(G) is exactly the set of irreducible Brauer characters at p.)From their definition, it is obvious that each partial character φ ∊ Iπ(G) can be “lifted” to an ordinary character χ ∊ Irr(G). (This means that φ is the restriction of χ to the π-elements of G.) In fact, there is a known set of canonical lifts Bπ(G) ⊆ Irr(G) for the irreducible π-partial characters. In this paper, it is proved that if 2 ∉ π, then there is an alternative set of canonical lifts (denoted Dπ(G)) that behaves better with respect to character induction.An application of this theory to M-groups is presented. If G is an M-group and S ⊆ G is a subnormal subgroup, consider a primitive character θ ⊆ Irr(S). It was known previously that if |G : S| is odd, then θ must be linear. It is proved here without restriction on the index of S that θ(1) is a power of 2.

scholarly journals NONVANISHING ELEMENTS FOR BRAUER CHARACTERS

2015 ◽  
Vol 102 (1) ◽  
pp. 96-107 ◽  
Author(s):  
SILVIO DOLFI ◽  
EMANUELE PACIFICI ◽  
LUCIA SANUS
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Regular Element ◽  
Solvable Groups ◽  
Vector Spaces ◽  
Brauer Character ◽  
Brauer Characters ◽  
A Finite Group ◽  
Finite Vector Spaces

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 [‘Zeros of Brauer characters and linear actions of finite groups’, J. Algebra 340 (2011), 104–113].

Brauer Characters of Prime Power Degrees and Conjugacy Classes of Prime Power Lengths

2010 ◽  
Vol 17 (04) ◽  
pp. 541-548 ◽  
Author(s):  
Pham Huu Tiep ◽  
Wolfgang Willems
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Prime Power ◽  
Conjugacy Classes ◽  
Brauer Character ◽  
Brauer Characters

We study the structure of any finite group in which the degree of every irreducible p-Brauer character, respectively the length of every p-regular conjugacy class, is a power of a fixed prime.

MONOLITHIC BRAUER CHARACTERS

2019 ◽  
Vol 100 (3) ◽  
pp. 434-439 ◽  
Author(s):  
XIAOYOU CHEN ◽  
MARK L. LEWIS
Keyword(s):  
Brauer Character ◽  
Brauer Characters

Let $G$ be a group, $p$ be a prime and $P\in \text{Syl}_{p}(G)$. We say that a $p$-Brauer character $\unicode[STIX]{x1D711}$ is monolithic if $G/\ker \unicode[STIX]{x1D711}$ is a monolith. We prove that $P$ is normal in $G$ if and only if $p\nmid \unicode[STIX]{x1D711}(1)$ for each monolithic Brauer character $\unicode[STIX]{x1D711}\in \text{IBr}(G)$. When $G$ is $p$-solvable, we also prove that $P$ is normal in $G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all monolithic irreducible $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.

Determination of Brauer Characters

1974 ◽  
Vol 26 (3) ◽  
pp. 746-752 ◽  
Author(s):  
B. M. Puttaswamaiah
Keyword(s):  
Group Algebra ◽  
Grothendieck Ring ◽  
Linear Character ◽  
Conjugate Class ◽  
Brauer Character ◽  
Integral Matrices ◽  
Brauer Characters ◽  
Characteristic Values ◽  
Category Of Modules

The purpose of this note is to show that the values of an irreducible (Brauer) character are the characteristic values of a matrix with non-negative rational integers. The construction of these integral matrices is done by a description of a representation of the Grothendieck ring of the category of modules over the group algebra. In particular a result of Solomon on characters and a result of Burnside on vanishing of a non-linear character on some conjugate class are generalized.

scholarly journals ITÔ’S THEOREM AND MONOMIAL BRAUER CHARACTERS

2017 ◽  
Vol 96 (3) ◽  
pp. 426-428 ◽  
Author(s):  
XIAOYOU CHEN ◽  
MARK L. LEWIS
Keyword(s):  
Solvable Group ◽  
Finite Solvable Group ◽  
Brauer Character ◽  
Brauer Characters

Let $G$ be a finite solvable group and let $p$ be a prime. In this note, we prove that $p$ does not divide $\unicode[STIX]{x1D711}(1)$ for every irreducible monomial $p$-Brauer character $\unicode[STIX]{x1D711}$ of $G$ if and only if $G$ has a normal Sylow $p$-subgroup.

Character Correspondences and π-Special Characters in π-Separable Groups

1987 ◽  
Vol 39 (4) ◽  
pp. 920-937 ◽  
Author(s):  
Thomas R. Wolf
Keyword(s):  
Irreducible Character ◽  
Solvable Groups ◽  
Character Theory ◽  
Separable Group ◽  
Irreducible Characters ◽  
Special Character ◽  
Brauer Characters ◽  
One To One

Let π be a set of primes and let G be a π-separable group (all groups considered are finite). Two subsets Xπ(G) and Bπ(G) of the set Irr(G) of irreducible characters of G play an important role in the character theory of π-separable groups and particularly solvable groups. If p is prime and π is the set of all other primes, then the Bπ characters of G give a natural one-to-one lift of the Brauer characters of G into Irr(G). More generally, they have been used to define Brauer characters for sets of primes.The π-special characters of G (i.e., Xπ(G)) restrict irreducibly and in a one-to-one fashion to a Hall-π-subgroup of G. If an irreducible character χ is quasi-primitive, it factors uniquely as a product of a π-special character an a π′-special character. This is a particularly useful tool in solvable groups.

Upper and Lower Bounds of Table Sums

2021 ◽  
Vol 28 (04) ◽  
pp. 555-560
Author(s):  
Xiaoyou Chen ◽  
Mark L. Lewis ◽  
Hung P. Tong-Viet
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Character Table ◽  
Upper And Lower Bounds ◽  
Separable Group ◽  
Brauer Character

For a group [Formula: see text], we produce upper and lower bounds for the sum of the entries of the Brauer character table of [Formula: see text] and the projective indecomposable character table of [Formula: see text]. When [Formula: see text] is a [Formula: see text]-separable group, we show that the sum of the entries in the table of Isaacs' partial characters is a real number, and we obtain upper and lower bounds for this sum.

The complements in an affine group

2013 ◽  
Vol 50 (2) ◽  
pp. 258-265
Author(s):  
Pál Hegedűs
Keyword(s):  
Permutation Group ◽  
Structural Similarity ◽  
Affine Group ◽  
Permutation Module ◽  
Regular Module ◽  
Brauer Characters

In this paper we analyse the natural permutation module of an affine permutation group. For this the regular module of an elementary Abelian p-group is described in detail. We consider the inequivalent permutation modules coming from nonconjugate complements. We prove their strong structural similarity well exceeding the fact that they have equal Brauer characters.

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