A note on maximal subgroups of σ-soluble groups

2021 ◽  
pp. 1-5
Author(s):  
Ning Su ◽  
Chenchen Cao ◽  
ShouHong Qiao
Keyword(s):  
Maximal Subgroups ◽  
Soluble Groups

A Note on the δ-length of Maximal Subgroups in Finite Soluble Groups

1994 ◽  
Vol 166 (1) ◽  
pp. 67-70 ◽  
Author(s):  
A. Ballester-Bolinches ◽  
M. D. Pérez-Ramos
Keyword(s):  
Maximal Subgroups ◽  
Soluble Groups

scholarly journals Schunck classes and projectors in a class of locally finite groups

1995 ◽  
Vol 38 (3) ◽  
pp. 511-522 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Groups ◽  
Maximal Subgroups ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Finite Groups ◽  
Finite Case ◽  
Saturated Formations ◽  
Definition Of ◽  
Schunck Class

We introduce a definition of a Schunck class of periodic abelian-by-finite soluble groups using major subgroups in place of the maximal subgroups used in Finite groups. This allows us to develop the theory as in the finite case proving the existence and conjugacy of projectors. Saturated formations are examples of Schunck classes and we are also able to obtain an infinite version of Gaschütz Ω-subgroups.

A note on minimal coverings of groups by subgroups

2001 ◽  
Vol 71 (2) ◽  
pp. 159-168 ◽  
Author(s):  
R. A. Bryce ◽  
L. Serena
Keyword(s):  
Maximal Subgroups ◽  
Soluble Groups ◽  
Finite Set ◽  
Group A ◽  
Abelian Subgroups

AbstractA cover for a group is a finite set of subgroups whose union is the whole group. A cover is minimal if its cardinality is minimal. Minimal covers of finite soluble groups are categorised; in particular all but at most one of their members are maximal subgroups. A characterisation is given of groups with minimal covers consisting of abelian subgroups.

Critical maximal subgroups and conjugacy of supplements in finite soluble groups

2018 ◽  
Vol 21 (1) ◽  
pp. 45-63
Author(s):  
Barbara Baumeister ◽  
Gil Kaplan
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Maximal Subgroups ◽  
Abelian Normal Subgroup ◽  
Soluble Groups ◽  
A Finite Group

AbstractLetGbe a finite group with an abelian normal subgroupN. When doesNhave a unique conjugacy class of complements inG? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups ofGclosed under conjugation whose intersection equals{\Phi(G)}. In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when{\Phi(G)=1}, these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.

On groups with countably many maximal subgroups

2018 ◽  
Vol 21 (2) ◽  
pp. 253-271 ◽  
Author(s):  
Ahmet Arikan ◽  
Giovanni Cutolo ◽  
Derek J. S. Robinson
Keyword(s):  
Maximal Subgroups ◽  
Group Rings ◽  
Soluble Groups

AbstractThe object of this work is to find classes of groups which possess only countably many maximal subgroups. Modules with countably many maximal submodules and group rings having countably many maximal right ideals are also investigated. Examples of soluble groups with uncountably many maximal subgroups are described.

Finite soluble groups with all n-maximal subgroups 𝔉${\mathfrak {F}}$ -subnormal

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
Viktoria A. Kovaleva ◽  
Alexander N. Skiba
Keyword(s):  
Maximal Subgroups ◽  
Soluble Groups

Abstract.We describe finite soluble groups in which every

scholarly journals Maximal subgroups and chief factors of certain generalized soluble groups

1971 ◽  
Vol 37 (2) ◽  
pp. 475-480 ◽  
Author(s):  
Richard Phillips ◽  
Derek Robinson ◽  
James Roseblade
Keyword(s):  
Maximal Subgroups ◽  
Soluble Groups ◽  
Chief Factors

scholarly journals On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2165
Author(s):  
Abd El-Rahman Heliel ◽  
Mohammed Al-Shomrani ◽  
Adolfo Ballester-Bolinches
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Direct Product ◽  
Prime Numbers ◽  
Chief Factor ◽  
Maximal Subgroups ◽  
Soluble Groups ◽  
Nilpotent Length ◽  
A Finite Group ◽  
Primary Groups

Let σ={σi:i∈I} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of |G| belong to the same member of σ. G is said to be σ-soluble if every chief factor of G is σ-primary, and G is σ-nilpotent if it is a direct product of σ-primary groups. It is known that G has a largest normal σ-nilpotent subgroup which is denoted by Fσ(G). Let n be a non-negative integer. The n-term of the σ-Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=Fσ(G/Fn(G)). If G is σ-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the σ-nilpotent length of G and it is denoted by lσ(G). If F is a subgroup-closed saturated formation, we define the σ-F-lengthnσ(G,F) of G as the σ-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a σ-soluble, then nσ(A,F)=nσ(G,F)−i for some i∈{0,1,2}.

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