scholarly journals Character degrees and CLT-groups

1989 ◽  
Vol 39 (2) ◽  
pp. 249-254 ◽  
Author(s):  
R. Brandl ◽  
P.A. Linnell
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Finite Groups ◽  
Character Degrees ◽  
A Finite Group

Let G be a finite group and let k be a field. We determine the smallest possible rank of a free kG-module that contains submodules of every possible dimension. As an application, we obtain various criteria for the wreath product of two finite groups to be a CLT-group.

Real class sizes and real character degrees

2010 ◽  
Vol 150 (1) ◽  
pp. 47-71 ◽  
Author(s):  
ROBERT M. GURALNICK ◽  
GABRIEL NAVARRO ◽  
PHAM HUU TIEP
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Character Degrees ◽  
Real Character ◽  
Fixed Field ◽  
Field Of Values ◽  
A Finite Group

Perhaps unexpectedly, there is a rich and deep connection between field of values of characters, their degrees and the structure of a finite group. Some of the fundamental results on the degrees of characters of finite groups, as the Ito–Michler and Thompson's theorems, admit a version involving only characters with certain fixed field of values ([DNT, NS, NST2, NT1, NT3]).

1-Connected character-graphs of finite groups with non-bipartite complement

2020 ◽  
pp. 2150058
Author(s):  
Mahdi Ebrahimi ◽  
Maryam khatami ◽  
Zohreh Mirzaei
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Character Degrees ◽  
Complex Character ◽  
Connected Graphs ◽  
A Finite Group

For a finite group [Formula: see text], let [Formula: see text] be the character-graph which is built on the set of irreducible complex character degrees of [Formula: see text]. In this paper, we wish to determine the structure of finite groups [Formula: see text] such that [Formula: see text] is 1-connected with nonbipartite complement. Also, we classify all 1-connected graphs with nonbipartite complement that can occur as the character-graph [Formula: see text] of a finite group [Formula: see text].

ON CONJUGACY CLASS SIZES AND CHARACTER DEGREES OF FINITE GROUPS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350100 ◽  
Author(s):  
GUOHUA QIAN ◽  
YANMING WANG
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Prime Power ◽  
P Element ◽  
Character Degrees ◽  
Prime Power Order ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Let p be a fixed prime, G a finite group and P a Sylow p-subgroup of G. The main results of this paper are as follows: (1) If gcd (p-1, |G|) = 1 and p2 does not divide |xG| for any p′-element x of prime power order, then G is a solvable p-nilpotent group and a Sylow p-subgroup of G/Op(G) is elementary abelian. (2) Suppose that G is p-solvable. If pp-1 does not divide |xG| for any element x of prime power order, then the p-length of G is at most one. (3) Suppose that G is p-solvable. If pp-1 does not divide χ(1) for any χ ∈ Irr (G), then both the p-length and p′-length of G are at most 2.

scholarly journals Irreducible characters of even degree and normal Sylow 2-subgroups

2016 ◽  
Vol 162 (2) ◽  
pp. 353-365 ◽  
Author(s):  
NGUYEN NGOC HUNG ◽  
PHAM HUU TIEP
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Average Degree ◽  
Character Degrees ◽  
Irreducible Characters ◽  
A Finite Group

AbstractThe classical Itô-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group G is coprime to a given prime p, then G has a normal Sylow p-subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of G is less than 4/3 then G has a normal Sylow 2-subgroup, as well as corresponding analogues for real-valued characters and strongly real characters. These results improve on several earlier results concerning the Itô-Michler theorem.

Robinson’s conjecture on heights of characters

2019 ◽  
Vol 155 (6) ◽  
pp. 1098-1117 ◽  
Author(s):  
Zhicheng Feng ◽  
Conghui Li ◽  
Yanjun Liu ◽  
Gunter Malle ◽  
Jiping Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Defect Group ◽  
Character Degrees ◽  
Simple Groups ◽  
Reductive Groups ◽  
A Finite Group

Geoffrey Robinson conjectured in 1996 that the $p$-part of character degrees in a $p$-block of a finite group can be bounded in terms of the center of a defect group of the block. We prove this conjecture for all primes $p\neq 2$ for all finite groups. Our argument relies on a reduction by Murai to the case of quasi-simple groups which are then studied using deep results on blocks of finite reductive groups.

scholarly journals ON THE REGULARITY CONJECTURE FOR THE COHOMOLOGY OF FINITE GROUPS

2008 ◽  
Vol 51 (2) ◽  
pp. 273-284 ◽  
Author(s):  
David J. Benson
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Finite Groups ◽  
Cohomology Ring ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Characteristic P ◽  
Residue Classes ◽  
The Difference ◽  
A Finite Group

AbstractLet $K$ be a field of characteristic $p$ and let $G$ be a finite group of order divisible by $p$. The regularity conjecture states that the Castelnuovo–Mumford regularity of the cohomology ring $H^*(G,K)$ is always equal to 0. We prove that if the regularity conjecture holds for a finite group $H$, then it holds for the wreath product $H\wr\mathbb{Z}/p$. As a corollary, we prove the regularity conjecture for the symmetric groups $\varSigma_n$. The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

Finite groups satisfying character degree congruences

2018 ◽  
Vol 21 (6) ◽  
pp. 1073-1094
Author(s):  
Peter Schmid
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Character Degree ◽  
Character Degrees ◽  
A Finite Group ◽  
Nonlinear Irreducible Character

Abstract Let G be a finite group, p a prime and {c\in\{0,1,\ldots,p-1\}} . Suppose that the degree of every nonlinear irreducible character of G is congruent to c modulo p. If here {c=0} , then G has a normal p-complement by a well known theorem of Thompson. We prove that in the cases where {c\neq 0} the group G is solvable with a normal abelian Sylow p-subgroup. If {p\neq 3} then this is true provided these character degrees are congruent to c or to {-c} modulo p.

Finite groups with two composite character degrees

2017 ◽  
Vol 16 (12) ◽  
pp. 1750228 ◽  
Author(s):  
Mehdi Ghaffarzadeh ◽  
Mohsen Ghasemi
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Character Degree ◽  
Character Degrees ◽  
A Finite Group

Let [Formula: see text] be a finite group and let [Formula: see text] be the set of all irreducible character degrees of [Formula: see text]. We consider finite groups [Formula: see text] with the property that [Formula: see text] has at most two composite members. We derive a bound 6 for the size of character degree sets of such groups. There are examples in both solvable and nonsolvable groups where this bound is met. In the case of nonsolvable groups, we are able to determine the structure of such groups with [Formula: see text].

scholarly journals ZL-amenability Constants of Finite Groups with Two Character Degrees

2014 ◽  
Vol 57 (3) ◽  
pp. 449-462 ◽  
Author(s):  
Mahmood Alaghmandan ◽  
Yemon Choi ◽  
Ebrahim Samei
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Character Degrees ◽  
The Third ◽  
A Finite Group

AbstractWe calculate the exact amenability constant of the centre of ℓ1(G) when G is a finite group and is either dihedral, extraspecial, or Frobenius with abelian complement and kernel. This is done using a formula that applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

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