scholarly journals Fitting classes and lattice formations I

2004 ◽  
Vol 76 (1) ◽  
pp. 93-108 ◽  
Author(s):  
M. Arroyo-Jordá ◽  
M. D. Pérez-Ramos
Keyword(s):  
Direct Product ◽  
Pairwise Disjoint ◽  
Nilpotent Groups ◽  
Saturated Formation ◽  
Closure Properties ◽  
Soluble Groups ◽  
Hall Subgroups ◽  
Disjoint Sets ◽  
Lattice Formation ◽  
Fitting Classes

AbstractA lattice formation is a class of groups whose elements are the direct product of Hall subgroups corresponding to pairwise disjoint sets of primes. In this paper Fitting classes with stronger closure properties involving F-subnormal subgroups, for a lattice formation F of full characteristic, are studied. For a subgroup-closed saturated formation G, a characterisation of the G-projectors of finite soluble groups is also obtained. It is inspired by the characterisation of the Carter subgroups as the N-projectors, N being the class of nilpotent groups.

scholarly journals Normal -Fitting classes

1992 ◽  
Vol 35 (2) ◽  
pp. 201-212
Author(s):  
J. C. Beidleman ◽  
M. J. Tomkinson
Keyword(s):  
Nilpotent Groups ◽  
Fitting Class ◽  
Soluble Groups ◽  
Locally Nilpotent ◽  
Normal Fitting Class ◽  
Fitting Classes

The authors together with M. J. Karbe [Ill. J. Math. 33 (1989) 333–359] have considered Fitting classes of -groups and, under some rather strong restrictions, obtained an existence and conjugacy theorem for -injectors. Results of Menegazzo and Newell show that these restrictions are, in fact, necessary.The Fitting class is normal if, for each is the unique -injector of G. is abelian normal if, for each. For finite soluble groups these two concepts coincide but the class of Černikov-by-nilpotent -groups is an example of a nonabelian normal Fitting class of -groups. In all known examples in which -injectors exist is closely associated with some normal Fitting class (the Černikov-by-nilpotent groups arise from studying the locally nilpotent injectors).Here we investigate normal Fitting classes further, paying particular attention to the distinctions between abelian and nonabelian normal Fitting classes. Products and intersections with (abelian) normal Fitting classes lead to further examples of Fitting classes satisfying the conditions of the existence and conjugacy theorem.

scholarly journals A Hall-type closure property for certain Fitting classes

1995 ◽  
Vol 59 (2) ◽  
pp. 204-213
Author(s):  
Owen J. Brison
Keyword(s):  
Closure Property ◽  
Closure Operation ◽  
Soluble Groups ◽  
Hall Subgroups ◽  
Fitting Classes

AbstractA closure operation connected with Hall subgroups is introduced for classes of finite soluble groups, and it is shown that this operation can be used to give a criterion for membership of certain special Fitting classes, including the so-called ‘central-socle’ classes.

scholarly journals Fitting classes and lattice formations II

2004 ◽  
Vol 76 (2) ◽  
pp. 175-188
Author(s):  
M. Arroyo-Jordá ◽  
M. D. Pérez-Ramos
Keyword(s):  
Fitting Class ◽  
Closure Properties ◽  
Lattice Formation ◽  
Fitting Classes ◽  
Good Behaviour ◽  
Subnormal Subgroups

AbstractGiven a lattice formation F of full characteristic, an F - Fitting class is a Fitting class with stronger closure properties involving F -subnormal subgroups. The main aim of this paper is to prove that the associated injectors possess a good behaviour with respect to F -subnormal subgroups.

An example in the theory of solubl groups

1970 ◽  
Vol 67 (1) ◽  
pp. 13-16 ◽  
Author(s):  
T. O. Hawkes
Keyword(s):  
Soluble Group ◽  
Nilpotent Groups ◽  
Saturated Formation ◽  
Finite Soluble Group ◽  
Carter Subgroup ◽  
Soluble Groups ◽  
University Of Wisconsin ◽  
Unpublished Work ◽  
The University

Let G be a finite soluble group. In (1) Alperin proves that two system normalizers of G contained in the same Carter subgroup C of G are conjugate in C. In recent unpublished work G.A.Chambers of the University of Wisconsin has proved that, if is a saturated formation, the -normalizers of an A-group are pronormal subgruops; hence, in particular, that two -normalizers contained in an -projector E of an A-group are conjugate in E. In this note we describe an example which shows that in Alperin's theorem the class of nilpotent groups cannot in general be replaced by an arbitary saturated formation without some restriction on the class of soluble groups under consideration. we provePROPOSITION. There exists a saturated formationand a group G which has two-normalizers E1and E2contained in an-projector F of G such that E1and E2are not conjugate in F.

Splitting theorems in abelian-by-hypercyclic groups

1978 ◽  
Vol 25 (1) ◽  
pp. 71-91 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Nilpotent Groups ◽  
Saturated Formation ◽  
Abelian Normal Subgroup ◽  
Soluble Groups ◽  
Rank Restrictions ◽  
A Finite Group ◽  
Special Case

AbstractIf is a saturated formation of finite soluble groups and G is a finite group whose -residual A is abelian then it is well known that G splits over A and the complements are conjugate. Hartley and Tomkinson (1975) considered the special case of this result in which is the class of nilpotent groups and obtained similar results for abelian-by-hypercentral groups with rank restrictions on the abelian normal subgroup. Here we consider the super-soluble case, obtaining corresponding results for abelian-by-hypercyclic groups.

scholarly journals A family of dominant Fitting classes of finite soluble groups

1998 ◽  
Vol 64 (1) ◽  
pp. 33-43 ◽  
Author(s):  
A. Ballester-Bolinches ◽  
A. Martínez-Pastor ◽  
M. D. Pérez-Ramos
Keyword(s):  
Large Family ◽  
Soluble Groups ◽  
Fitting Classes

AbstractIn this paper a large family of dominant Fitting classes of finite soluble groups and the description of the corresponding injectors are obtained. Classical constructions of nilpotent and Lockett injectors as well as p-nilpotent injectors arise as particular cases.

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 Maximal Fitting classes of finite soluble groups

1974 ◽  
Vol 10 (2) ◽  
pp. 169-175 ◽  
Author(s):  
R.A. Bryce ◽  
John Cossey
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fitting Class ◽  
Soluble Groups ◽  
Necessary And Sufficient ◽  
Fitting Classes

From recent results of Lausch, it is easy to establish necessary and sufficient conditions for a Fitting class to be maximal in the class of all finite soluble groups. We use Lausch's methods to show that there are normal Fitting classes not contained in any Fitting class maximal in the class of all finite soluble groups. We also find conditions on Fitting classes and for to be maximal in .

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

Sign in / Sign up

Export Citation Format

Share Document