On a class of finite soluble groups

2018 ◽  
Vol 21 (5) ◽  
pp. 839-846 ◽  
Author(s):  
Adolfo Ballester-Bolinches ◽  
John Cossey ◽  
Yangming Li
Keyword(s):  
Finite Groups ◽  
Soluble Groups ◽  
Closed Formation

Abstract The aim of this paper is to study the class of finite groups in which every subgroup is self-normalising in its subnormal closure. It is proved that this class is a subgroup-closed formation of finite soluble groups which is not closed under taking Frattini extensions and whose members can be characterised by means of their Carter subgroups. This leads to new characterisations of finite soluble T-, PT- and PST-groups. Finite groups whose p-subgroups, p a prime, are self-normalising in their subnormal closure are also characterised.

scholarly journals Quasinormal Fitting classes of finite groups

2019 ◽  
pp. 18-26
Author(s):  
Martsinkevich Anna V.
Keyword(s):  
Natural Number ◽  
Wreath Product ◽  
Cyclic Group ◽  
Finite Groups ◽  
Fitting Class ◽  
Soluble Groups ◽  
Normal Fitting Class ◽  
Fitting Classes ◽  
Local Fitting ◽  
Class F

Let P be the set of all primes, Zn a cyclic group of order n and X wr Zn the regular wreath product of the group X with Zn. A Fitting class F is said to be X-quasinormal (or quasinormal in a class of groups X ) if F ⊆ X, p is a prime, groups G ∈ F and G wr Zp ∈ X, then there exists a natural number m such that G m wr Zp ∈ F. If  X is the class of all soluble groups, then F is normal Fitting class. In this paper we generalize the well-known theorem of Blessenohl and Gaschütz in the theory of normal Fitting classes. It is proved, that the intersection of any set of nontrivial X-quasinormal Fitting classes is a nontrivial X-quasinormal Fitting class. In particular, there exists the smallest nontrivial X-quasinormal Fitting class. We confirm a generalized version of the Lockett conjecture (in particular, the Lockett conjecture) about the structure of a Fitting class for the case of X-quasinormal classes, where X is a local Fitting class of partially 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.

ON SYLOW NORMALIZERS OF FINITE GROUPS

2013 ◽  
Vol 13 (03) ◽  
pp. 1350116 ◽  
Author(s):  
L. S. KAZARIN ◽  
A. MARTÍNEZ-PASTOR ◽  
M. D. PÉREZ-RAMOS
Keyword(s):  
Finite Groups ◽  
Property A ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Hall Subgroups ◽  
Saturated Formations ◽  
The Universe ◽  
Do So

The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

FC-nilpotent and FC-soluble groups

1956 ◽  
Vol 52 (3) ◽  
pp. 391-398 ◽  
Author(s):  
A. M. Duguid ◽  
D. H. McLain
Keyword(s):  
Finite Number ◽  
Finite Groups ◽  
Abelian Groups ◽  
Characteristic Subgroup ◽  
Arbitrary Group ◽  
Soluble Groups ◽  
Image Position

Let an element of a group be called an FC element if it has only a finite number of conjugates in the group. Baer(1) and Neumann (8) have discussed groups in which every element is FC, and called them FC-groups. Both Abelian and finite groups are trivially FC-groups; Neumann has studied the properties common to FC-groups and Abelian groups, and Baer the properties common to FC-groups and finite groups. Baer has also shown that, for an arbitrary group G, the set H1 of all FC elements is a characteristic subgroup. Haimo (3) has defined the FC-chain of a group G byHi/Hi−1 is the subgroup of all FC elements in G/Hi−1.

Formation theory and groups of automorphisms of -groups

1977 ◽  
Vol 76 (4) ◽  
pp. 255-265 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Groups ◽  
Composition Series ◽  
Formation Theory ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Finite Groups ◽  
Group Of Automorphisms ◽  
Infinite Case ◽  
Groups Of Automorphisms ◽  
Automorphisms Of Groups

SynopsisFurther results from the theory of finite soluble groups are extended to the class of locally finite groups with a satisfactory Sylow structure. Let be a saturated U-formation and A a -group of automorphisms of the -group G. A is said to act -centrally on G if G has an A-composition series (Λσ/Vσ; σ ∈ ∑) such that A induces an f(p)-group of automorphisms in each p-factor Λσ/Vσ. We show that in this situation A is an -group, thus generalising the result of Schmid [8]. Associated results of Schmid and of Baer are also extended to the infinite case.

Some Sylow subgroups

1961 ◽  
Vol 260 (1302) ◽  
pp. 304-316 ◽  
Keyword(s):  
Finite Groups ◽  
Infinite Groups ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Uncountable Group

Most of the well-known theorems of Sylow for finite groups and of P. Hall for finite soluble groups have been extended to certain restricted classes of infinite groups. To show the limitations of such generalizations, examples are here constructed of infinite groups subject to stringent but natural restrictions, groups in which certain Sylow or Hall theorems fail. All the groups are metabelian and of exponent 6. There are countable such groups in which a Sylow 2-subgroup has a complement but no Sylow 3-complement; or again no complement at all. There are countable such groups with continuously many mutually non-isomorphic Sylow 3-subgroups. There are groups, necessarily of uncountable order, with Sylow 2-subgroups of different orders. The most elaborate example is of an uncountable group in which all Sylow 2-subgroups and 3-subgroups are countable, and none is complemented.

Finite quasisimple groups of 2×2 matrices over a division ring

1985 ◽  
Vol 97 (3) ◽  
pp. 415-420
Author(s):  
B. Hartley ◽  
M. A. Shahabi Shojaei
Keyword(s):  
Finite Groups ◽  
Division Ring ◽  
Characteristic Zero ◽  
Multiplicative Group ◽  
Main Tool ◽  
Number Fields ◽  
Soluble Groups ◽  
Trivial Element ◽  
Difficult Part

In 1955 [1], Amitsur determined all the finite groups G that can be embedded in the multiplicative group T* = GL(1, T) of some division ring T of characteristic zero. If G can be so embedded, then the rational span of G in T is a division ring of finite dimension over ℚ, and G acts on it by right multiplication in such a way that every non-trivial element operates fixed point freely. The finite groups admitting such a representation had earlier been determined by Zassenhaus[24; 4, XII. 8], and Amitsur begins by quoting Zassenhaus' results, which show in particular that the only perfect group that can be embedded in the multiplicative group of a division ring of characteristic zero is SL(2,5). The more difficult part of Amitsur's paper is the determination of the possible soluble groups. Here the main tool is Hasse's theory of cyclic algebras over number fields.

scholarly journals Local definitions of local homomorphs and formations of finite groups

1985 ◽  
Vol 31 (1) ◽  
pp. 5-34 ◽  
Author(s):  
P. Förster ◽  
E. Salomon
Keyword(s):  
Finite Groups ◽  
Local Formation ◽  
Soluble Groups ◽  
Formation Of Finite Groups ◽  
Formation Function ◽  
Definition Of

It is well known that every local formation of finite soluble groups possesses three distinguished local definitions consisting of finite soluble groups: the minimal one, the full and integrated one, and the maximal one. As far as the first and the second of these are concerned, this statement remains true in the context of arbitrary finite groups. Doerk, Šemetkov, and Schmid have posed the problem of whether every local formation of finite groups has a distinguished (that is, unique) maximal local definition. In this paper a description of local formations with a unique maximal local definition is given, from which counter-examples emerge. Furthermore, a criterion for a formation function to be a local definition of a given local formation is obtained.

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