scholarly journals On Groups Whose Irreducible Character Degrees of All Proper Subgroups are All Prime Powers

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Shitian Liu
Keyword(s):  
Finite Groups ◽  
Irreducible Character ◽  
Prime Power ◽  
Character Degree ◽  
Character Degrees ◽  
Simple Groups ◽  
Nonsolvable Group ◽  
Irreducible Characters ◽  
Almost Simple Groups

Isaacs, Passman, and Manz have determined the structure of finite groups whose each degree of the irreducible characters is a prime power. In particular, if G is a nonsolvable group and every character degree of a group G is a prime power, then G is isomorphic to S × A , where S ∈ A 5 , PSL 2 8 and A is abelian. In this paper, we change the condition, each character degree of a group G is a prime power, into the condition, each character degree of the proper subgroups of a group is a prime power, and give the structure of almost simple groups whose character degrees of all proper subgroups are all prime powers.

Coprime Group Actions Fixing All Nonlinear Irreducible Characters

1989 ◽  
Vol 41 (1) ◽  
pp. 68-82 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Finite Groups ◽  
Irreducible Character ◽  
Group Actions ◽  
Character Degrees ◽  
Prime Divisors ◽  
Irreducible Characters ◽  
Nilpotent Length ◽  
Derived Subgroup ◽  
Nonlinear Irreducible Character

The main result of this paper is the following:Theorem A. Let H and N be finite groups with coprime orders andsuppose that H acts nontrivially on N via automorphisms. Assume that Hfixes every nonlinear irreducible character of N. Then the derived subgroup ofN is nilpotent and so N is solvable of nilpotent length≦ 2.Why might one be interested in a situation like this? There has been considerable interest in the question of what one can deduce about a group Gfrom a knowledge of the setcd(G) = ﹛x(l)lx ∈ Irr(G) ﹜of irreducible character degrees of G.Recently, attention has been focused on the prime divisors of the elements of cd(G). For instance, in [9], O. Manz and R. Staszewski consider π-separable groups (for some set π of primes) with the property that every element of cd(G) is either a 77-number or a π'-number.

Finite groups with at most six vanishing conjugacy classes

2021 ◽  
pp. 2250076
Author(s):  
Sajjad M. Robati ◽  
M. R. Darafsheh
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Conjugacy Classes ◽  
Character Formula ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
A Finite Group

Let [Formula: see text] be a finite group. We say that a conjugacy class of [Formula: see text] in [Formula: see text] is vanishing if there exists some irreducible character [Formula: see text] of [Formula: see text] such that [Formula: see text]. In this paper, we show that finite groups with at most six vanishing conjugacy classes are solvable or almost simple groups.

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].

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.

GROUPS WITH THE SAME CHARACTER DEGREES AS SPORADIC ALMOST SIMPLE GROUPS

2016 ◽  
Vol 94 (2) ◽  
pp. 254-265
Author(s):  
SEYED HASSAN ALAVI ◽  
ASHRAF DANESHKHAH ◽  
ALI JAFARI
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Irreducible Character ◽  
Abelian Subgroup ◽  
Character Degrees ◽  
Simple Groups ◽  
Sporadic Simple Group ◽  
Almost Simple Group ◽  
Almost Simple Groups ◽  
A Finite Group

Let$G$be a finite group and$\mathsf{cd}(G)$denote the set of complex irreducible character degrees of$G$. We prove that if$G$is a finite group and$H$is an almost simple group whose socle is a sporadic simple group$H_{0}$and such that$\mathsf{cd}(G)=\mathsf{cd}(H)$, then$G^{\prime }\cong H_{0}$and there exists an abelian subgroup$A$of$G$such that$G/A$is isomorphic to$H$. In view of Huppert’s conjecture, we also provide some examples to show that$G$is not necessarily a direct product of$A$and$H$, so that we cannot extend the conjecture to almost simple groups.

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.

Non-solvable groups each of whose character degrees has at most two prime divisors

2020 ◽  
pp. 2150030 ◽  
Author(s):  
Babak Miraali ◽  
Sajjad M. Robati
Keyword(s):  
Simple Group ◽  
Solvable Group ◽  
Solvable Groups ◽  
Character Degree ◽  
Character Degrees ◽  
Simple Groups ◽  
Prime Divisors ◽  
Almost Simple Group ◽  
Almost Simple Groups

In this paper, we determine all almost simple groups each of whose character degrees has at most two distinct prime divisors. More generally, we show that a finite non-solvable group [Formula: see text] with this property is an extension of an almost simple group [Formula: see text] by a solvable group and [Formula: see text], where [Formula: see text] is the set of all primes dividing some character degree of [Formula: see text].

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].

NONDIVISIBILITY AMONG IRREDUCIBLE CHARACTER CO-DEGREES

2021 ◽  
pp. 1-7
Author(s):  
NEDA AHANJIDEH
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Irreducible Characters ◽  
Formula Group ◽  
A Finite Group

Abstract For a character $\chi $ of a finite group G, the number $\chi ^c(1)={[G:{\textrm {ker}}\chi ]}/{\chi (1)}$ is called the co-degree of $\chi $ . A finite group G is an ${\textrm {NDAC}} $ -group (no divisibility among co-degrees) when $\chi ^c(1) \nmid \phi ^c(1)$ for all irreducible characters $\chi $ and $\phi $ of G with $1< \chi ^c(1) < \phi ^c(1)$ . We study finite groups admitting an irreducible character whose co-degree is a given prime p and finite nonsolvable ${\textrm {NDAC}} $ -groups. Then we show that the finite simple groups $^2B_2(2^{2f+1})$ , where $f\geq 1$ , $\mbox {PSL}_3(4)$ , ${\textrm {Alt}}_7$ and $J_1$ are determined uniquely by the set of their irreducible character co-degrees.

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