scholarly journals Noncommutative rational functions invariant under the action of a finite solvable group

2020 ◽  
Vol 490 (2) ◽  
pp. 124341
Author(s):  
Igor Klep ◽  
James Eldred Pascoe ◽  
Gregor Podlogar ◽  
Jurij Volčič
Keyword(s):  
Solvable Group ◽  
Rational Functions ◽  
Finite Solvable Group

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.

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

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.

Sylow Normalizers with a Normal Sylow 2-Subgroup

2008 ◽  
Vol 51 (3) ◽  
pp. 779-783 ◽  
Author(s):  
Gabriel Navarro ◽  
Lucía Sanus
Keyword(s):  
Solvable Group ◽  
Finite Solvable Group ◽  
Brauer Characters

AbstractIf G is a finite solvable group and p is a prime, then the normalizer of a Sylow p-subgroup has a normal Sylow 2-subgroup if and only if all non-trivial irreducible real 2-Brauer characters of G have degree divisible by p.

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 /.

Numbers Of Conjugacy Class Sizes And Derived Lengths for A-Groups

1996 ◽  
Vol 39 (3) ◽  
pp. 346-351 ◽  
Author(s):  
Mary K. Marshall
Keyword(s):  
Conjugacy Class ◽  
Solvable Group ◽  
Conjugacy Classes ◽  
Finite Solvable Group ◽  
Conjugacy Class Sizes ◽  
Sylow Subgroups ◽  
Derived Length ◽  
Completely Reducible

AbstractAn A-group is a finite solvable group all of whose Sylow subgroups are abelian. In this paper, we are interested in bounding the derived length of an A-group G as a function of the number of distinct sizes of the conjugacy classes of G. Although we do not find a specific bound of this type, we do prove that such a bound exists. We also prove that if G is an A-group with a faithful and completely reducible G-module V, then the derived length of G is bounded by a function of the number of distinct orbit sizes under the action of G on V.

C2-Cofiniteness of Cyclic-Orbifold Vertex Operator Superalgebras

2017 ◽  
Vol 24 (02) ◽  
pp. 315-322
Author(s):  
Chunrui Ai
Keyword(s):  
Fixed Point ◽  
Solvable Group ◽  
Vertex Operator ◽  
Finite Solvable Group ◽  
Vertex Operator Superalgebra

In this paper, it is proved that for any C2-cofinite, simple, CFT-type vertex operator superalgebra V and a finite solvable group G consisting of automorphisms of V, the fixed point subalgebra VG is a C2-cofinite vertex operator superalgebra.

Groups of automorphisms and centralizers

1990 ◽  
Vol 107 (2) ◽  
pp. 227-238 ◽  
Author(s):  
Alexandre Turull
Keyword(s):  
Solvable Group ◽  
Finite Solvable Group ◽  
Group Of Automorphisms ◽  
Fitting Height ◽  
Groups Of Automorphisms ◽  
The Difference

Let G be a finite solvable group and A a group of automorphisms of G such that (|A|, |G|) = 1. We denote by h(G) the Fitting height of G and by l(A) the length of the longest chain of subgroups of A. Then, under some additional hypotheses, it is known from [5] that h(G) ≤ 2l(A) + h(CG(A)) and from [8] that, when CG(A) = 1, h(G) ≤ l(A), both results being best possible (see [6, 7]). The present paper attempts to explain the difference in the coefficient of l(A) in the two inequalities, from 2 to 1.

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