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.

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.

More soluble groups with primitive group rings

1983 ◽  
Vol 41 (1) ◽  
pp. 1-4
Author(s):  
C. J. B. Brookes
Keyword(s):  
Primitive Group ◽  
Group Rings ◽  
Soluble Groups

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.

ON MODULES OVER GROUP RINGS OF LOCALLY SOLUBLE GROUPS WITH RANK RESTRICTIONS ON SOME SYSTEMS OF SUBGROUPS

2010 ◽  
Vol 03 (01) ◽  
pp. 45-55 ◽  
Author(s):  
O. Yu. Dashkova
Keyword(s):  
Soluble Group ◽  
Proper Subgroup ◽  
Dedekind Domain ◽  
Group Rings ◽  
Soluble Groups ◽  
Rank Restrictions ◽  
Locally Soluble Group ◽  
Dg Module

We consider a DG-module A over a Dedekind domain D. Let G be a group having infinite section p-rank (or infinite 0-rank) such that CG(A) = 1. It is known that if A is not an artinian D-module then for every proper subgroup H, the quotient A/CA(H) is an artinian D-module for every proper subgroup H of infinite section p-rank (or infinite 0-rank respectively). In this paper, it is proved that if G is a locally soluble group, then G is soluble. Some properties of soluble groups of this type are also obtained.

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

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.

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