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 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.

Finite Solvable Groups Whose Character Degree Graphs Are Not Complete

2006 ◽  
Vol 13 (04) ◽  
pp. 541-552 ◽  
Author(s):  
Jiping Zhang
Keyword(s):  
Prime Number ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Nilpotent Subgroup ◽  
Character Degree ◽  
Finite Solvable Group

In this paper, we characterize the finite solvable groups with non-complete character degree graphs by proving the following theorem, which generalizes a conjecture by Huppert. Suppose that G is a finite solvable group and p is a prime number dividing the degree of some irreducible character of G. If there is another such prime number q such that pq does not divide the degree of any irreducible character of G, then both p-length ℓp(G) and q-length ℓq(G) of G are at most two, and ℓp(G)+ ℓq(G)=4 if and only if pq=6 with QG/Zφ(QG)≅ 32:GL(2,3), where QG is generated by all Sylow 2-subgroups of G and Zφ(G) is a normal nilpotent subgroup of G. Moreover, the bounds are best possible.

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.

On normal subgroups of M-groups

2019 ◽  
Vol 18 (04) ◽  
pp. 1950074
Author(s):  
Xuewu Chang
Keyword(s):  
Normal Subgroup ◽  
Solvable Group ◽  
Solvable Groups ◽  
Hall Subgroup ◽  
Embedding Theorem ◽  
The Other ◽  
Finite Solvable Group ◽  
Embedding Problem ◽  
Normal Subgroups ◽  
Other Hand

The normal embedding problem of finite solvable groups into [Formula: see text]-groups was studied. It was proved that for a finite solvable group [Formula: see text], if [Formula: see text] has a special normal nilpotent Hall subgroup, then [Formula: see text] cannot be a normal subgroup of any [Formula: see text]-group; on the other hand, if [Formula: see text] has a maximal normal subgroup which is an [Formula: see text]-group, then [Formula: see text] can occur as a normal subgroup of an [Formula: see text]-group under some suitable conditions. The results generalize the normal embedding theorem on solvable minimal non-[Formula: see text]-groups to arbitrary [Formula: see text]-groups due to van der Waall, and also cover the famous counterexample given by Dade and van der Waall independently to the Dornhoff’s conjecture which states that normal subgroups of arbitrary [Formula: see text]-groups must be [Formula: see text]-groups.

A Note on Brauer Character Degrees of Solvable Groups

1991 ◽  
Vol 34 (3) ◽  
pp. 423-425 ◽  
Author(s):  
You-Qiang Wang
Keyword(s):  
Solvable Group ◽  
Solvable Groups ◽  
Character Degrees ◽  
Finite Solvable Group ◽  
Prime Integer ◽  
Derived Length ◽  
Brauer Character ◽  
Brauer Characters

AbstractLet G be a finite solvable group. Fix a prime integer p and let t be the number of distinct degrees of irreducible Brauer characters of G with respect to the prime p. We obtain the bound 3t — 2 for the derived length of a Hall p'-subgroup of G. Furthermore, if |G| is odd, then the derived length of a Hall p'-subgroup of G is bounded by /.

Character correspondences in solvable groups with a self-normalizing sylow subgroup

2019 ◽  
Vol 19 (10) ◽  
pp. 2050190
Author(s):  
Carolina Vallejo Rodríguez
Keyword(s):  
Solvable Group ◽  
Irreducible Character ◽  
Sylow Subgroup ◽  
Solvable Groups ◽  
Character Formula ◽  
Finite Solvable Group ◽  
Irreducible Characters ◽  
Made In

Let [Formula: see text] be a finite solvable group and let [Formula: see text] for some prime [Formula: see text]. Whenever [Formula: see text] is odd, Isaacs described a correspondence between irreducible characters of degree not divisible by [Formula: see text] of [Formula: see text] and [Formula: see text]. This correspondence is natural in the sense that an algorithm is provided to compute it, and the result of the application of the algorithm does not depend on choices made. In the case where [Formula: see text], G. Navarro showed that every irreducible character [Formula: see text] of degree not divisible by [Formula: see text] has a unique linear constituent [Formula: see text] when restricted to [Formula: see text], and that the map [Formula: see text] defines a bijection. Navarro’s bijection is obviously natural in the sense described above. We show that these two correspondences are the same under the intersection of the hypotheses.

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.

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).

Prime Factors of π-Partial Character Degrees and Conjugacy Class Sizes of π-Elements

2005 ◽  
Vol 12 (04) ◽  
pp. 699-707
Author(s):  
Antonio Beltrán ◽  
María José Felipe
Keyword(s):  
Conjugacy Class ◽  
Solvable Group ◽  
Character Degree ◽  
Character Degrees ◽  
Finite Solvable Group ◽  
Conjugacy Class Sizes ◽  
Prime Factors

Let G be a finite solvable group. We prove that any prime dividing any irreducible π-partial character degree of G divides the size of some conjugacy class of π-elements of G. Under certain hypothesis, we show that if two distinct primes r and s both divide some irreducible π-partial character degree, then there exists a conjugacy class of π-elements whose size is divisible by rs.

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