scholarly journals Solvable groups whose character degree graphs generalize squares

2020 ◽  
Vol 23 (2) ◽  
pp. 217-234
Author(s):  
Mark L. Lewis ◽  
Qingyun Meng
Keyword(s):  
Solvable Group ◽  
Solvable Groups ◽  
Character Degree ◽  
Sylow Subgroups ◽  
Character Degree Graph ◽  
Product Of Subgroups ◽  
Degree Graph ◽  
Delta G ◽  
Definition Of ◽  
Square Graph

AbstractLet G be a solvable group, and let {\Delta(G)} be the character degree graph of G. In this paper, we generalize the definition of a square graph to graphs that are block squares. We show that if G is a solvable group so that {\Delta(G)} is a block square, then G has at most two normal nonabelian Sylow subgroups. Furthermore, we show that when G is a solvable group that has two normal nonabelian Sylow subgroups and {\Delta(G)} is block square, then G is a direct product of subgroups having disconnected character degree graphs.

On the character degree graph of solvable groups. II. Discon- nected graphs

2001 ◽  
Vol 38 (1-4) ◽  
pp. 339-355 ◽  
Author(s):  
P. P. Pálfy
Keyword(s):  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Character Degree ◽  
Finite Solvable Group ◽  
Nilpotent Length ◽  
Character Degree Graph ◽  
Degree Graph

Let G be a finite solvable group and ¡ a set of primes such that the degree of each irreducible character of G is either a ¡-number or a ¡0-number. We show that such groups have a very restricted structure, for example, their nilpotent length is at most 4. We also prove that Huppert's ^{ÿ Conjecture is valid for these groups.

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 character degree graph of solvable groups

2017 ◽  
Vol 146 (4) ◽  
pp. 1505-1513 ◽  
Author(s):  
Zeinab Akhlaghi ◽  
Carlo Casolo ◽  
Silvio Dolfi ◽  
Khatoon Khedri ◽  
Emanuele Pacifici
Keyword(s):  
Solvable Groups ◽  
Character Degree ◽  
Character Degree Graph ◽  
Degree Graph

scholarly journals Locally defined Fitting classes

1975 ◽  
Vol 20 (1) ◽  
pp. 25-32 ◽  
Author(s):  
Patrick D' Arcy
Keyword(s):  
Conjugacy Class ◽  
Solvable Group ◽  
Solvable Groups ◽  
Fitting Class ◽  
Finite Solvable Group ◽  
Saturated Formations ◽  
Definition Of ◽  
Image Position ◽  
Fitting Classes

Fitting classes of finite solvable groups were first considered by Fischer, who with Gäschutz and Hartley (1967) showed in that in each finite solvable group there is a unique conjugacy class of “-injectors”, for a Fitting class. In general the behaviour of Fitting classes and injectors seems somewhat mysterious and hard to determine. This is in contrast to the situation for saturated formations and -projectors of finite solvable groups which, because of the equivalence saturated formations and locally defined formations, can be studied in a much more detailed way. However for those Fitting classes that are “locally defined” the theory of -injectors can be made more explicit by considering various centralizers involving the local definition of , giving results analogous to some of those concerning locally defined formations. Particular attention will be given to the subgroup B() defined by where the set {(p)} of Fitting classes locally defines , and the Sp are the Sylow p-subgroups associated with a given Sylow system − B() plays a role very much like that of Graddon's -reducer in Graddon (1971). An -injector of B() is an -injector of G, and for certain simple B() is an -injector of G.

MONOMIAL AND MONOLITHIC CHARACTERS OF FINITE SOLVABLE GROUPS

2021 ◽  
pp. 1-9
Author(s):  
BURCU ÇINARCI
Keyword(s):  
Solvable Group ◽  
Prime Divisor ◽  
Solvable Groups ◽  
Character Degree ◽  
Finite Solvable Group

Abstract Let G be a finite solvable group and let p be a prime divisor of $|G|$ . We prove that if every monomial monolithic character degree of G is divisible by p, then G has a normal p-complement and, if p is relatively prime to every monomial monolithic character degree of G, then G has a normal Sylow p-subgroup. We also classify all finite solvable groups having a unique imprimitive monolithic character.

On permutable strongly 2-maximal and strongly 3-maximal subgroups

2021 ◽  
pp. 121-129
Author(s):  
Yuliya V. Gorbatova
Keyword(s):  
Nilpotent Group ◽  
Solvable Group ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Maximal Subgroups ◽  
Finite Solvable Group ◽  
Prime Divisors ◽  
Sylow Subgroups ◽  
Schmidt Group ◽  
Number Of Prime Divisors

We describe the structure of finite solvable non-nilpotent groups in which every two strongly n-maximal subgroups are permutable (n = 2; 3). In particular, it is shown for a solvable non-nilpotent group G that any two strongly 2-maximal subgroups are permutable if and only if G is a Schmidt group with Abelian Sylow subgroups. We also prove the equivalence of the structure of non-nilpotent solvable groups with permutable 3-maximal subgroups and with permutable strongly 3-maximal subgroups. The last result allows us to classify all finite solvable groups with permutable strongly 3-maximal subgroups, and we describe 14 classes of groups with this property. The obtained results also prove the nilpotency of a finite solvable group with permutable strongly n -maximal subgroups if the number of prime divisors of the order of this group strictly exceeds n (n=2; 3).

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 On the character degree graph of finite groups

2019 ◽  
Vol 198 (5) ◽  
pp. 1595-1614 ◽  
Author(s):  
Zeinab Akhlaghi ◽  
Carlo Casolo ◽  
Silvio Dolfi ◽  
Emanuele Pacifici ◽  
Lucia Sanus
Keyword(s):  
Finite Groups ◽  
Character Degree ◽  
Character Degree Graph ◽  
Degree Graph
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