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$M_{p}$ -GROUPS AND BRAUER CHARACTER DEGREES

Authors

Xiaoyou Chen, Mark L. Lewis
Type: Article
Publication Date: 2025-05-09
Citations: 0
DOI: https://doi.org/10.1017/s0004972725000280

Abstract

Abstract Let G be a finite group and p be a prime. We prove that if G has three codegrees, then G is an M -group. We prove for some prime p that if the degree of every nonlinear irreducible Brauer character of G is a prime, then for every normal subgroup N of G , either $G/N$ or N is an $M_p$ -group.

Locations

  • Bulletin of the Australian Mathematical Society
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References (13)

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1998-05-07
Gabriel Navarro
This is a clear, accessible and up-to-date exposition of modular representation theory of finite groups from a character-theoretic viewpoint. After a short review of the necessary background material, the early … This is a clear, accessible and up-to-date exposition of modular representation theory of finite groups from a character-theoretic viewpoint. After a short review of the necessary background material, the early chapters introduce Brauer characters and blocks and develop their basic properties. The next three chapters study and prove Brauer's first, second and third main theorems in turn. These results are then applied to prove a major application of finite groups, the Glauberman Z*-theorem. Later chapters examine Brauer characters in more detail. The relationship between blocks and normal subgroups is also explored and the modular characters and blocks in p-solvable groups are discussed. Finally, the character theory of groups with a Sylow p-subgroup of order p is studied. Each chapter concludes with a set of problems. The book is aimed at graduate students, with some previous knowledge of ordinary character theory, and researchers studying the representation theory of finite groups.

Finite groups whose irreducible Brauer characters have prime power degrees

2014-06-24
Hung P. Tong‐Viet
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1991-06-01
David Chillag , Avinoam Mann , Olaf Manz

Zeros of Brauer characters

2012-06-07
Huiqun Wang , Xiaoyou Chen , Jiwen Zeng

Module correspondence in finite groups

1981-02-01
Tetsuro Okuyama
In section 5, we try to give some refinement of Brauer's induction theorem in terms of p -blocks.Dress's induction theorem is also considered there.Let G be a finite group and … In section 5, we try to give some refinement of Brauer's induction theorem in terms of p -blocks.Dress's induction theorem is also considered there.Let G be a finite group and p a fixed rational prime number.In this paper we use the following notations and terminologies and for other notations and terminologies we shall refer to books of Gorenstein [16], Dornhoff [8] and Feit [11].R ; a complete discrete valuation ring with maximal ideal (\pi)\ni p .K ; the quotient field of R. F;=R/(\pi) , the residue field of R which has characteristic; the group algebras of G over R, K and F respec- tively.We assume that fields K and F are both splitting fields for all groups considered in this paper.All modules considered are right unital and finitely generated.V^{N} ; the induced module for an F[H] -module V to N where H and N are subgroups of G with H\subseteqq N .V_{1H} ; the restriction of an F[N] -module V to H where H and N are subgroups of G with H\subseteqq N .V^{x} ; the conjugate module of an F[H] -module V by an element x in G which is an F[H^{x}] -module.If H is a normal subgroup of G, we define the inertia subgroup of V, denoted by I_{G}(V) , to be the set \{x\in G|V^{x}\cong V\} .V|W ; for F[G] modules V and W, V is isomorphic to a direct summand of W.In the above we use the same notations for R[T] -modules too for a subgroup T of G. L_{0}(G) ; an F[G] -module of dimension 1 on which G acts trivially.For F[G] modules V and W and a subgroup H of G define ( V, W)_{H}=Hom_{F[H]} ( V, W) .The relative norm map T_{H,G} ; ( V, W)_{H}arrow(V, W)_{G} is defined as follows.T_{H,G}( \lambda)=\sum_{x}\lambda^{x} for \lambda\in(V, W)_{H} , where x ranges over a set of representatives of right cosets of H in G and \lambda^{x} is the map varrow(\lambda(vx^{-1}))x .If \mathfrak{H}_{\vee} is a set of subgroups of G, we define (V, W)_{G}^{\mathfrak{H}}=(V, W)_{G}/(V, W)_{\mathfrak{H},G} where (V, W)_{\mathfrak{H},G}= \sum_{H\in \mathfrak{H}} Im T_{H,G} .Now we shall describe the Green correspondence which is frequently used in the paper.We refer to a book of Feit [11].Let V be an indecomposable F[G] -module.Then there exists a psubgroup D which is a minimal subgroup of G such that V is F[D] -projec- tive.A group D is uniquely determined up to conjugation by an element
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Co-degrees of irreducible characters in finite groups

2006-12-13
Guohua Qian , Yanming Wang , Huaquan Wei

An answer to two questions of Brewster and Yeh on M-groups

2003-07-16
Alexander Moretó

Codegrees and nilpotence class of <i>p</i>-groups

2015-12-17
Ni Du , Mark L. Lewis
Abstract If χ is an irreducible character of a finite group G , then the codegree of χ is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>|</m:mo> <m:mi>G</m:mi> <m:mo>:</m:mo> <m:mi>ker</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>χ</m:mi> <m:mo>)</m:mo> … Abstract If χ is an irreducible character of a finite group G , then the codegree of χ is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>|</m:mo> <m:mi>G</m:mi> <m:mo>:</m:mo> <m:mi>ker</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>χ</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>|</m:mo> <m:mo>/</m:mo> <m:mi>χ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${|G:{\rm ker}(\chi)|/\chi(1)}$ . We show that if G is a p -group, then the nilpotence class of G is bounded in terms of the largest codegree for an irreducible character of G .

The analog of Huppert's conjecture on character codegrees

2016-12-30
Yong Yang , Guohua Qian

Groups with few codegrees of irreducible characters

2019-01-20
Fereydoon Alizadeh , Houshang Behravesh , Mehdi Ghaffarzadeh +2 more
For a character χ of a finite group G, the number cod(χ)=|G:Ker(χ)|/χ(1) is called the codegree of χ. We also define cod(G)={cod(χ)∣χ∈Irr(G)}. In this article, we give a characterization of … For a character χ of a finite group G, the number cod(χ)=|G:Ker(χ)|/χ(1) is called the codegree of χ. We also define cod(G)={cod(χ)∣χ∈Irr(G)}. In this article, we give a characterization of finite groups with at most three codegrees of irreducible characters.

Character degrees, character codegrees and nilpotence class of <i>p</i>-groups

2021-09-09
Alexander Moretó
Du and Lewis raised in 2016 the question of whether the nilpotence class of a p-group is bounded in terms of the number of character codegrees. In 2020, Croome and … Du and Lewis raised in 2016 the question of whether the nilpotence class of a p-group is bounded in terms of the number of character codegrees. In 2020, Croome and Lewis, gave a positive answer to this question for p-groups with four character codegree under some additional hypotheses related, for instance, to the number of character degrees of the group. In this note, we show that in general the nilpotence class of a p-group is bounded in terms of the number of character degrees and the number of character codegrees. In the case of four character codegrees, we extend some of the results of Croome and Lewis.

A characterization of groups in terms of the degrees of their characters. II

1968-03-01
I. M. Isaacs , D. S. Passman
In this paper we continue our study of the relationship between the structure of a finite group G and the set of degrees of its irreducible complex characters.The following hypotheses … In this paper we continue our study of the relationship between the structure of a finite group G and the set of degrees of its irreducible complex characters.The following hypotheses on the degrees are considered: (A) G has r.x. e for some prime p, i.e. all the degrees divide p e , (B) the degrees are linearly ordered by divisibility and all except 1 are divisible by exactly the same set of primes, (C) G has a.c.m, i.e., all the degrees except 1 are equal to some fixed m, (D) all the degrees except 1 are prime (not necessarily the same prime) and (E) all the degrees except 1 are divisible by p e > p but none is divisible by p e+1 .In each of these situations, group theoretic information is deduced from the character theoretic hypothesis and in several cases complete characterizations are obtained.
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Character Theory of Finite Groups

1976-01-01
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