Fitting classes and injectors

2006 ◽  
pp. 309-354
Keyword(s):  
Fitting Classes

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 .

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.

A problem in the theory of normal fitting classes

1975 ◽  
Vol 141 (2) ◽  
pp. 99-110 ◽  
Author(s):  
R. A. Bryce ◽  
John Cossey
Keyword(s):  
Fitting Classes

On σ-local Fitting classes

2020 ◽  
Vol 542 ◽  
pp. 116-129 ◽  
Author(s):  
Wenbin Guo ◽  
Li Zhang ◽  
N.T. Vorob'ev
Keyword(s):  
Fitting Classes ◽  
Local Fitting

Hall operators for Fitting classes

1979 ◽  
Vol 33 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Owen J. Brison
Keyword(s):  
Fitting Classes

scholarly journals Locally defined Fitting classes

1975 ◽  
Vol 20 (1) ◽  
pp. 25-32 ◽  
Author(s):  
Patrick D' Arcy
Keyword(s):  
Conjugacy Class ◽  
Solvable Group ◽  
Solvable Groups ◽  
Fitting Class ◽  
Finite Solvable Group ◽  
Saturated Formations ◽  
Definition Of ◽  
Image Position ◽  
Fitting Classes

Fitting classes of finite solvable groups were first considered by Fischer, who with Gäschutz and Hartley (1967) showed in that in each finite solvable group there is a unique conjugacy class of “-injectors”, for a Fitting class. In general the behaviour of Fitting classes and injectors seems somewhat mysterious and hard to determine. This is in contrast to the situation for saturated formations and -projectors of finite solvable groups which, because of the equivalence saturated formations and locally defined formations, can be studied in a much more detailed way. However for those Fitting classes that are “locally defined” the theory of -injectors can be made more explicit by considering various centralizers involving the local definition of , giving results analogous to some of those concerning locally defined formations. Particular attention will be given to the subgroup B() defined by where the set {(p)} of Fitting classes locally defines , and the Sp are the Sylow p-subgroups associated with a given Sylow system − B() plays a role very much like that of Graddon's -reducer in Graddon (1971). An -injector of B() is an -injector of G, and for certain simple B() is an -injector of G.

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