Construction of a Non-Solvable Group of Exponent 5

1973 ◽  
pp. 39-66 ◽  
Author(s):  
S. Bachmuth ◽  
H.Y. Mochizuki ◽  
D.W. Walkup
Keyword(s):  
Solvable Group

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].

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.

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.

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