scholarly journals Notes on the largest irreducible character degree of a finite group

2004 ◽  
Vol 132 (7) ◽  
pp. 1899-1903 ◽  
Author(s):  
Qian Guohua
Keyword(s):  
Finite Group ◽  
Irreducible Character ◽  
Character Degree ◽  
A Finite Group ◽  
Irreducible Character Degree

The largest Irreducible Character Degree of a Finite Group

1985 ◽  
Vol 37 (3) ◽  
pp. 442-451 ◽  
Author(s):  
David Gluck
Keyword(s):  
Finite Group ◽  
Irreducible Character ◽  
Abelian Subgroup ◽  
Character Degree ◽  
Character Table ◽  
Significant Information ◽  
Famous Theorem ◽  
Frobenius Reciprocity ◽  
A Finite Group ◽  
Irreducible Character Degree

Much information about a finite group is encoded in its character table. Indeed even a small portion of the character table may reveal significant information about the group. By a famous theorem of Jordan, knowing the degree of one faithful irreducible character of a finite group gives an upper bound for the index of its largest normal abelian subgroup.Here we consider b(G), the largest irreducible character degree of the group G. A simple application of Frobenius reciprocity shows that b(G) ≧ |G:A| for any abelian subgroup A of G. In light of this fact and Jordan's theorem, one might seek to bound the index of the largest abelian subgroup of G from above by a function of b(G). If is G is nilpotent, a result of Isaacs and Passman (see [7, Theorem 12.26]) shows that G has an abelian subgroup of index at most b(G)4.

scholarly journals A CHARACTERIZATION OF PSL(4,p) BY SOME CHARACTER DEGREE

2019 ◽  
pp. 679
Author(s):  
Younes Rezayi ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Irreducible Character ◽  
Character Degree ◽  
A Finite Group ◽  
Irreducible Character Degree

‎Let G be a finite group and cd(G) be the set of irreducible character degree of G‎. ‎In this paper we prove that if  p is a prime number‎, ‎then the simple group PSL(4,p) is uniquely determined by its order and some its character degrees‎. 

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 𝑝-closedness and 𝑝-nilpotency of finite groups

2019 ◽  
Vol 22 (5) ◽  
pp. 927-932
Author(s):  
Shuqin Dong ◽  
Hongfei Pan ◽  
Long Miao
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Class Size ◽  
Character Degree ◽  
Conjugacy Class Size ◽  
A Finite Group ◽  
Irreducible Character Degree

Abstract Let {\operatorname{acd}(G)} and {\operatorname{acs}(G)} denote the average irreducible character degree and the average conjugacy class size, respectively, of a finite group G. The object of this paper is to prove that if \operatorname{acd}(G)<2(p+1)/(p+3) , then G=O_{p}(G)\times O_{{p^{\prime}}}(G) , and that if \operatorname{acs}(G)<4p/(p\kern-1.0pt+\kern-1.0pt3) , then G=O_{p}(G)\kern-1.0pt\times\kern-1.0ptO_{{p^{\prime}}}(G) with {O_{p}(G)} abelian, where p is a prime.

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.

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.

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 𝑝-groups with exactly four codegrees

2020 ◽  
Vol 23 (6) ◽  
pp. 1111-1122
Author(s):  
Sarah Croome ◽  
Mark L. Lewis
Keyword(s):  
Irreducible Character ◽  
Character Degree ◽  
Character Degrees ◽  
Additional Hypothesis ◽  
Nilpotence Class ◽  
Irreducible Character Degree

AbstractLet G be a p-group, and let χ be an irreducible character of G. The codegree of χ is given by {\lvert G:\operatorname{ker}(\chi)\rvert/\chi(1)}. Du and Lewis have shown that a p-group with exactly three codegrees has nilpotence class at most 2. Here we investigate p-groups with exactly four codegrees. If, in addition to having exactly four codegrees, G has two irreducible character degrees, G has largest irreducible character degree {p^{2}}, {\lvert G:G^{\prime}\rvert=p^{2}}, or G has coclass at most 3, then G has nilpotence class at most 4. In the case of coclass at most 3, the order of G is bounded by {p^{7}}. With an additional hypothesis, we can extend this result to p-groups with four codegrees and coclass at most 6. In this case, the order of G is bounded by {p^{10}}.

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