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.

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.

Some character degree sets imply direct product

2016 ◽  
Vol 15 (10) ◽  
pp. 1650186
Author(s):  
Kamal Aziziheris ◽  
Heidar Sahrayi
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Direct Product ◽  
Group Structure ◽  
Character Degree ◽  
Character Degrees ◽  
Normal Subgroups ◽  
Arbitrary Positive Integer ◽  
A Finite Group ◽  
Complex Characters

Let [Formula: see text] be the set of all irreducible complex characters of a finite group [Formula: see text]. In [K. Aziziheris, Determining group structure from sets of irreducible character degrees, J. Algebra 323 (2010) 1765–1782], we proved that if [Formula: see text] and [Formula: see text] are relatively prime integers greater than [Formula: see text], [Formula: see text] is prime not dividing [Formula: see text], and [Formula: see text] is a solvable group such that [Formula: see text], then under some conditions on [Formula: see text] and [Formula: see text], the group [Formula: see text] is the direct product of two normal subgroups, where [Formula: see text] and [Formula: see text]. In this paper, we replace [Formula: see text] by [Formula: see text], where [Formula: see text] is an arbitrary positive integer, and we obtain similar result. As an application, we show that if [Formula: see text] is a finite group with [Formula: see text] or [Formula: see text], then [Formula: see text] is a direct product of two non-abelian normal subgroups.

Bounding Fitting Heights of Two Classes of Character Degree Graphs

2014 ◽  
Vol 21 (02) ◽  
pp. 355-360
Author(s):  
Xianxiu Zhang ◽  
Guangxiang Zhang
Keyword(s):  
Solvable Group ◽  
Disjoint Union ◽  
Character Degree ◽  
Finite Solvable Group ◽  
Total Degree ◽  
Character Degree Graph ◽  
Vertex Set ◽  
Fitting Height ◽  
Degree Graph

In this article, we prove that a finite solvable group with character degree graph containing at least four vertices has Fitting height at most 4 if each derived subgraph of four vertices has total degree not more than 8. We also prove that if the vertex set ρ(G) of the character degree graph Δ(G) of a solvable group G is a disjoint union ρ(G) = π1 ∪ π2, where |πi| ≥ 2 and pi, qi∈ πi for i = 1,2, and no vertex in π1 is adjacent in Δ(G) to any vertex in π2 except for p1p2 and q1q2, then the Fitting height of G is at most 4.

scholarly journals On the minimum degree of the power graph of a finite cyclic group

2020 ◽  
pp. 2150044
Author(s):  
Ramesh Prasad Panda ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Finite Groups ◽  
Minimum Degree ◽  
Simple Graph ◽  
The Other ◽  
Prime Divisors ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The power graph [Formula: see text] of a finite group [Formula: see text] is the undirected simple graph whose vertex set is [Formula: see text], in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer [Formula: see text], let [Formula: see text] denote the cyclic group of order [Formula: see text] and let [Formula: see text] be the number of distinct prime divisors of [Formula: see text]. The minimum degree [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For [Formula: see text], under certain conditions involving the prime divisors of [Formula: see text], we identify at most [Formula: see text] vertices such that [Formula: see text] is equal to the degree of at least one of these vertices. If [Formula: see text], or that [Formula: see text] is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of [Formula: see text].

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].

N-critical graph of finite groups

2021 ◽  
Author(s):  
Viachaslau I. Murashka
Keyword(s):  
Finite Group ◽  
Directed Graph ◽  
Finite Groups ◽  
Soluble Group ◽  
Prime Divisors ◽  
Nilpotent Length ◽  
Critical Graph ◽  
Vertex Set ◽  
A Finite Group

A Schmidt [Formula: see text]-group is a non-nilpotent [Formula: see text]-group whose proper subgroups are nilpotent and which has the normal Sylow [Formula: see text]-subgroup. The [Formula: see text]-critical graph [Formula: see text] of a finite group [Formula: see text] is a directed graph on the vertex set [Formula: see text] of all prime divisors of [Formula: see text] and [Formula: see text] is an edge of [Formula: see text] if and only if [Formula: see text] has a Schmidt [Formula: see text]-subgroup. The bounds of the nilpotent length of a soluble group are obtained in terms of its [Formula: see text]-critical graph. The structure of a soluble group with given [Formula: see text]-critical graph is obtained in terms of commutators. The connections between [Formula: see text]-critical and other graphs (Sylow, soluble, prime, commuting) of finite groups are found.

scholarly journals On Thompson’s Conjecture for Alternating GroupsAp+3

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Shitian Liu ◽  
Yong Yang
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Conjugacy Classes ◽  
Connected Components ◽  
Prime Divisors ◽  
Vertex Set ◽  
Thompson’S Conjecture ◽  
Trivial Center ◽  
A Finite Group

LetGbe a group. Denote byπ(G)the set of prime divisors of|G|. LetGK(G)be the graph with vertex setπ(G)such that two primespandqinπ(G)are joined by an edge ifGhas an element of orderp·q. We sets(G)to denote the number of connected components of the prime graphGK(G). Denote byN(G)the set of nonidentity orders of conjugacy classes of elements inG. Alavi and Daneshkhah proved that the groups,Anwheren=p,p+1,p+2withs(G)≥2, are characterized byN(G). As a development of these topics, we will prove that ifGis a finite group with trivial center andN(G)=N(Ap+3)withp+2composite, thenGis isomorphic toAp+3.

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.

A NOTE ON -PARTS OF BRAUER CHARACTER DEGREES

2020 ◽  
pp. 1-5
Author(s):  
JINBAO LI ◽  
YONG YANG
Keyword(s):  
Finite Group ◽  
Character Degree ◽  
Character Degrees ◽  
Brauer Character ◽  
A Finite Group

Let $G$ be a finite group and $p$ be an odd prime. We show that if $\mathbf{O}_{p}(G)=1$ and $p^{2}$ does not divide every irreducible $p$ -Brauer character degree of $G$ , then $|G|_{p}$ is bounded by $p^{3}$ when $p\geqslant 5$ or $p=3$ and $\mathsf{A}_{7}$ is not involved in $G$ , and by $3^{4}$ if $p=3$ and $\mathsf{A}_{7}$ is involved in $G$ .

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