On the Carter subgroup of a solvable group

1969 ◽  
Vol 109 (4) ◽  
pp. 288-310 ◽  
Author(s):  
Jack Shamash
Keyword(s):  
Solvable Group ◽  
Carter Subgroup

scholarly journals Frobenius action on Carter subgroups

2020 ◽  
Vol 30 (05) ◽  
pp. 1073-1080
Author(s):  
Güli̇n Ercan ◽  
İsmai̇l Ş. Güloğlu
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Semidirect Product ◽  
Frobenius Group ◽  
Product Formula ◽  
Finite Solvable Group ◽  
Carter Subgroup ◽  
Text Length ◽  
Fitting Height ◽  
A Finite Group

Let [Formula: see text] be a finite solvable group and [Formula: see text] be a subgroup of [Formula: see text]. Suppose that there exists an [Formula: see text]-invariant Carter subgroup [Formula: see text] of [Formula: see text] such that the semidirect product [Formula: see text] is a Frobenius group with kernel [Formula: see text] and complement [Formula: see text]. We prove that the terms of the Fitting series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the Fitting series of [Formula: see text], and the Fitting height of [Formula: see text] may exceed the Fitting height of [Formula: see text] by at most one. As a corollary it is shown that for any set of primes [Formula: see text], the terms of the [Formula: see text]-series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the [Formula: see text]-series of [Formula: see text], and the [Formula: see text]-length of [Formula: see text] may exceed the [Formula: see text]-length of [Formula: see text] by at most one. These theorems generalize the main results in [E. I. Khukhro, Fitting height of a finite group with a Frobenius group of automorphisms, J. Algebra 366 (2012) 1–11] obtained by Khukhro.

Sous-groupes de Carter dans les groupes de rang de Morley fini

2004 ◽  
Vol 69 (1) ◽  
pp. 23-33
Author(s):  
Olivier Frécon
Keyword(s):  
Conjugacy Class ◽  
Solvable Group ◽  
Closed Subgroup ◽  
Nilpotent Subgroup ◽  
Morley Rank ◽  
Carter Subgroup ◽  
Locally Nilpotent ◽  
Finite Morley Rank

RésuméA Carter subgroup is a self-normalizing locally nilpotent subgroup. For studying these subgroups in groups of finite Morley rank, we introduce the new notion of a locally closed subgroup. We show that every solvable group of finite Morley rank has a unique conjugacy class of Carter subgroups.

𝒫-characters and the structure of finite solvable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiakuan Lu ◽  
Kaisun Wu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Irreducible Constituent ◽  
A Finite Group

AbstractLet 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

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.

Group Algebras with Minimal Strong Lie Derived Length

2008 ◽  
Vol 51 (2) ◽  
pp. 291-297 ◽  
Author(s):  
Ernesto Spinelli
Keyword(s):  
Solvable Group ◽  
Group Algebra ◽  
Positive Characteristic ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Algebras ◽  
Derived Length ◽  
Necessary And Sufficient

AbstractLet KG be a non-commutative strongly Lie solvable group algebra of a group G over a field K of positive characteristic p. In this note we state necessary and sufficient conditions so that the strong Lie derived length of KG assumes its minimal value, namely [log2(p + 1)].

ITÔ’S THEOREM AND MONOMIAL BRAUER CHARACTERS II

2017 ◽  
Vol 97 (2) ◽  
pp. 215-217
Author(s):  
XIAOYOU CHEN ◽  
MARK L. LEWIS
Keyword(s):  
Solvable Group ◽  
Finite Solvable Group ◽  
Brauer Characters

Let $G$ be a finite solvable group and let $p$ be a prime. We prove that the intersection of the kernels of irreducible monomial $p$-Brauer characters of $G$ with degrees divisible by $p$ is $p$-closed.

Real characters, monolithic characters and the Taketa inequality

2019 ◽  
Vol 18 (10) ◽  
pp. 1950183 ◽  
Author(s):  
Burcu Çınarcı ◽  
Temha Erkoç
Keyword(s):  
Solvable Group ◽  
Character Degrees ◽  
Finite Solvable Group ◽  
Complex Character ◽  
The Real ◽  
Derived Length

In this paper, we prove that the Taketa inequality, namely the derived length of a finite solvable group [Formula: see text] is less than or equal to the number of distinct irreducible complex character degrees of [Formula: see text], is true under some conditions related to the real and the monolithic characters of [Formula: see text].

Ranks of the factors of the lower central series for a free solvable group

1973 ◽  
Vol 13 (3) ◽  
pp. 488-492
Author(s):  
G. P. Egorychev
Keyword(s):  
Solvable Group ◽  
Lower Central Series ◽  
Central Series ◽  
Free Solvable Group

The 2-length and 2-period of a finite solvable group

1979 ◽  
Vol 18 (1) ◽  
pp. 5-20 ◽  
Author(s):  
E. G. Bryukhanova
Keyword(s):  
Solvable Group ◽  
Finite Solvable Group

Derived length when a product of characters has two nonprincipal irreducible constituents

2016 ◽  
Vol 15 (06) ◽  
pp. 1650110
Author(s):  
Lisa Rose Hendrixson ◽  
Mark L. Lewis
Keyword(s):  
Solvable Group ◽  
Irreducible Character ◽  
Character Formula ◽  
Derived Length

We study the situation where a solvable group [Formula: see text] has a faithful irreducible character [Formula: see text] such that [Formula: see text] has exactly two distinct nonprincipal irreducible constituents. We prove that [Formula: see text] has derived length bounded above by 8, and provide an example of such a group having derived length 8. In particular, this improves upon a result of Adan-Bante.

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