Character Degree Graphs of Solvable Groups of Fitting Height 2

2006 ◽  
Vol 49 (1) ◽  
pp. 127-133 ◽  
Author(s):  
Mark L. Lewis
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Character Degree ◽  
Character Degrees ◽  
Irreducible Characters ◽  
Vertex Set ◽  
Fitting Height ◽  
A Finite Group

AbstractGiven a finite group G, we attach to the character degrees of G a graph whose vertex set is the set of primes dividing the degrees of irreducible characters of G, and with an edge between p and q if pq divides the degree of some irreducible character of G. In this paper, we describe which graphs occur when G is a solvable group of Fitting height 2.

Schur indices and irreducible character degrees in finite solvable groups

1989 ◽  
Vol 105 (2) ◽  
pp. 237-240 ◽  
Author(s):  
R. Gow
Keyword(s):  
Finite Group ◽  
Irreducible Character ◽  
Group Structure ◽  
Solvable Groups ◽  
Character Degrees ◽  
Irreducible Characters ◽  
A Finite Group

Let G be a finite group and let Irr(G) denote the set of complex irreducible characters of G. Various authors have investigated the question of how information about the degrees of the characters in Irr (G) can provide information about the structure of G. Chapter 12 of [2] gives a survey of a number of results arising from such questions. Two well-known examples of theorems that relate character degrees and group structure are those due to Thompson (12·2 in [2]) and Itô (12.34 in [2]), which we recall here.

𝒫-characters and the structure of finite solvable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiakuan Lu ◽  
Kaisun Wu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Irreducible Constituent ◽  
A Finite Group

AbstractLet 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

scholarly journals A Note on a Conjecture of Brauer

1963 ◽  
Vol 22 ◽  
pp. 1-13 ◽  
Author(s):  
Paul Fong
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Solvable Group ◽  
Prime Divisor ◽  
Irreducible Character ◽  
Defect Group ◽  
Irreducible Characters ◽  
A Finite Group

In [1] R. Brauer asked the following question: Let be a finite group, p a rational prime number, and B a p-block of with defect d and defect group . Is it true that is abelian if and only if every irreducible character in B has height 0 ? The present results on this problem are quite incomplete. If d-0, 1, 2 the conjecture was proved by Brauer and Feit, [4] Theorem 2. They also showed that if is cyclic, then no characters of positive height appear in B. If is normal in , the conjecture was proved by W. Reynolds and M. Suzuki, [12]. In this paper we shall show that for a solvable group , the conjecture is true for the largest prime divisor p of the order of . Actually, one half of this has already been proved in [7]. There it was shown that if is a p-solvable group, where p is any prime, and if is abelian, then the condition on the irreducible characters in B is satisfied.

BOUNDING THE NUMBER OF IRREDUCIBLE CHARACTER DEGREES OF A FINITE GROUP IN TERMS OF THE LARGEST DEGREE

2013 ◽  
Vol 13 (02) ◽  
pp. 1350096 ◽  
Author(s):  
MARK L. LEWIS ◽  
ALEXANDER MORETÓ
Keyword(s):  
Finite Group ◽  
Irreducible Character ◽  
Character Degree ◽  
Character Degrees ◽  
Character Degree Graph ◽  
Prime Factors ◽  
Degree Graph ◽  
A Finite Group

We conjecture that the number of irreducible character degrees of a finite group is bounded in terms of the number of prime factors (counting multiplicities) of the largest character degree. We prove that this conjecture holds when the largest character degree is prime and when the character degree graph is disconnected.

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.

The solvability comes from a given set of character degrees

2016 ◽  
Vol 15 (09) ◽  
pp. 1650164 ◽  
Author(s):  
Farideh Shafiei ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Irreducible Character ◽  
Prime Power ◽  
Character Degree ◽  
Character Degrees ◽  
Degree Set ◽  
A Finite Group ◽  
Irreducible Character Degree

Let [Formula: see text] be a finite group and the irreducible character degree set of [Formula: see text] is contained in [Formula: see text], where [Formula: see text], and [Formula: see text] are distinct integers. We show that one of the following statements holds: [Formula: see text] is solvable; [Formula: see text]; or [Formula: see text] for some prime power [Formula: see text].

On the Largest Irreducible Character Degree of a Finite Solvable Group

2010 ◽  
Vol 17 (spec01) ◽  
pp. 925-927 ◽  
Author(s):  
M. H. Jafari
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Solvable Group ◽  
Irreducible Character ◽  
Prime Order ◽  
Character Degree ◽  
Finite Solvable Group ◽  
A Finite Group ◽  
Irreducible Character Degree

Let b(G) denote the largest irreducible character degree of a finite group G. In this paper, we prove that if G is a solvable group which does not involve a section isomorphic to the wreath product of two groups of equal prime order p, and if b(G) < pn, then |G:Op(G)|p < pn.

ON THE REGULARITY OF CHARACTER DEGREE GRAPHS

2019 ◽  
Vol 100 (3) ◽  
pp. 428-433 ◽  
Author(s):  
Z. SAYANJALI ◽  
Z. AKHLAGHI ◽  
B. KHOSRAVI
Keyword(s):  
Finite Group ◽  
Character Degree ◽  
Character Degrees ◽  
Prime Divisors ◽  
Character Degree Graph ◽  
Vertex Set ◽  
Degree Graph ◽  
A Finite Group ◽  
Regular Bipartite Graph ◽  
Complex Characters

Let $G$ be a finite group and let $\text{Irr}(G)$ be the set of all irreducible complex characters of $G$. Let $\unicode[STIX]{x1D70C}(G)$ be the set of all prime divisors of character degrees of $G$. The character degree graph $\unicode[STIX]{x1D6E5}(G)$ associated to $G$ is a graph whose vertex set is $\unicode[STIX]{x1D70C}(G)$, and there is an edge between two distinct primes $p$ and $q$ if and only if $pq$ divides $\unicode[STIX]{x1D712}(1)$ for some $\unicode[STIX]{x1D712}\in \text{Irr}(G)$. We prove that $\unicode[STIX]{x1D6E5}(G)$ is $k$-regular for some natural number $k$ if and only if $\overline{\unicode[STIX]{x1D6E5}}(G)$ is a regular bipartite graph.

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 Whose Common-Divisor Graph is Regular

2018 ◽  
Vol 61 (2) ◽  
pp. 329-341
Author(s):  
Mehdi Ghaffarzadeh ◽  
Mohsen Ghasemi ◽  
Mark L. Lewis
Keyword(s):  
Finite Group ◽  
Regular Graph ◽  
Finite Groups ◽  
Complete Graph ◽  
Solvable Groups ◽  
Connected Components ◽  
Irreducible Characters ◽  
Vertex Set ◽  
The Common ◽  
A Finite Group

AbstractLet G be a finite group, and write cd (G) for the set of degrees of irreducible characters of G. The common-divisor graph Γ(G) associated with G is the graph whose vertex set is cd (G)∖{1} and there is an edge between distinct vertices a and b, if (a, b) > 1. In this paper we prove that if Γ(G) is a k-regular graph for some k ⩾ 0, then for the solvable groups, either Γ(G) is a complete graph of order k + 1 or Γ(G) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd(G) = cd(PSL2(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).

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