Subgroup closed Fitting classes are formations

1982 ◽  
Vol 91 (2) ◽  
pp. 225-258 ◽  
Author(s):  
R. A. Bryce ◽  
John Cossey
Keyword(s):  
Fitting Class ◽  
The Other ◽  
Soluble Groups ◽  
Intrinsic Interest ◽  
Nilpotent Length ◽  
Good Account ◽  
The One ◽  
Second Category ◽  
Fitting Classes

Since their introduction by Fischer(12) and Fischer, Gaschütz and Hartley (13) Fitting classes of soluble groups have attracted attention on two fronts (all groups considered in this paper will be finite and soluble). On the one hand is their important role in the structure of finite soluble groups, a good account of which can be found in Gaschütz (14), and on the other is their intrinsic interest as classes of groups. This paper falls into the second category, and is a continuation and completion of (8). There we proved that a subgroup closed Fitting class is a formation if it consists of groups of nilpotent length at most three. Happily, at last, we can remove this qualification.

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.

scholarly journals A criterion for the Hall-closure of Fitting classes

1981 ◽  
Vol 23 (3) ◽  
pp. 361-365 ◽  
Author(s):  
Owen J. Brison
Keyword(s):  
Fitting Class ◽  
Soluble Groups ◽  
Normal Fitting Class ◽  
Fitting Classes

In a recent paper, Cusack has given a criterion, in terms of the Fitting class “join” operation, for a normal Fitting class to be closed under the taking of Hall π-subgroups. Here we show that Cusack's result can be slightly modified so as to give a criterion for any Fitting class of finite soluble groups to be closed under taking Hall π-subgroups.

scholarly journals Hall-closure and products of Fitting classes

1982 ◽  
Vol 32 (2) ◽  
pp. 145-164 ◽  
Author(s):  
Owen J. Brison
Keyword(s):  
Fitting Class ◽  
Soluble Groups ◽  
Normal Fitting Class ◽  
Group Construction ◽  
Case Part ◽  
Fitting Classes

AbstractThe Fitting class (of finite, soluble, groups), , is said to be Hall π-closed (where π is a set of primes) if whenever G is a group in and H is a Hall π-subgroup of G, then H belongs to . In this paper, we study the Hall π-closure of products of Fitting classes. Our main result is a characterisation of the Hall π-closedFitting classes of the form (where denotes the so-called smallest normal Fitting class), subject to a restriction connecting π with the characteristic of . We also characterise those Fitting classes (respectively, ) such that (respectively, ) is Hall π-closed for all Fitting classes . In each case, part of the proof uses a concrete group construction. As a bonus, one of these construction also yields a “cancellation result” for certain products of Fitting classes.

scholarly journals On Fitting classes that are Schunck classes

1981 ◽  
Vol 30 (3) ◽  
pp. 381-384 ◽  
Author(s):  
John Cossey
Keyword(s):  
Fitting Class ◽  
The Novel ◽  
Soluble Groups ◽  
Chief Factors ◽  
Fitting Classes ◽  
Schunck Class

AbstractAn example is given to show that a class of finite soluble groups that is both a Fitting class and a Schunck class need not be a formation. The novel feature of this class is that it is defined by imposing conditions on complemented chief factors of groups in it: this technique usually does not give rise to Fitting classes that are not formations.

scholarly journals On Local Embedding Properties of Injectors of Finite Soluble Groups

2004 ◽  
Vol 76 (1) ◽  
pp. 23-38
Author(s):  
Stephanie Reifferscheid
Keyword(s):  
Embedded System ◽  
Fitting Class ◽  
Permutable Subgroup ◽  
Soluble Groups ◽  
Fitting Classes

AbstractIn the present paper we consider Fitting classes of finite soluble groups which locally satisfy additional conditions related to the behaviour of their injectors. More precisely, we study Fitting classes 1 ≠⊆such that an-injector of G is, respectively, a normal, (sub)modular, normally embedded, system permutable subgroup of G for all G ∈.Locally normal Fitting classes were studied before by various authors. Here we prove that some important results—already known for normality—are valid for all of the above mentioned embedding properties. For instance, all these embedding properties behave nicely with respect to the Lockett section. Further, for all of these properties the class of all finite soluble groups G such that an x-injector of G has the corresponding embedding property is not closed under forming normal products, and thus can fail to be a Fitting class.

scholarly journals Quotient closed metanilpotent Fitting classes

1985 ◽  
Vol 38 (2) ◽  
pp. 157-163 ◽  
Author(s):  
T. R. Berger ◽  
R. A. Bryce ◽  
John Cossey
Keyword(s):  
Quotient Group ◽  
Fitting Class ◽  
Normal Subgroups ◽  
Soluble Groups ◽  
Fitting Classes

AbstractA Fitting class of finite soluble groups is one closed under the formation of normal subgroups and products of normal subgroups. It is shown that the Fitting classes of metanilpotent groups which are quotient group closed as well are primitive saturated formuations.

scholarly journals Fitting classes based on groups of nilpotent length three with operator-isomorphic minimal normal subgroups

1991 ◽  
Vol 51 (3) ◽  
pp. 448-467
Author(s):  
Brendan McCann
Keyword(s):  
Fitting Class ◽  
Normal Subgroups ◽  
Nilpotent Length ◽  
Fitting Classes

AbstractIn this paper a technique for constructing Fitting Classes is applied to certain groups of nilpotent length three which have non-unique minimal normal subgroups. A characterisation of the minimal Fitting Class of some of these groups is also given.

scholarly journals A note on injectors in finite soluble groups

1977 ◽  
Vol 17 (3) ◽  
pp. 419-421 ◽  
Author(s):  
John Cossey
Keyword(s):  
Fitting Class ◽  
Soluble Groups ◽  
Nilpotent Length ◽  
Chief Factors ◽  
Phd Thesis

Groups of nilpotent length four containing a subgroup which covers and avoids the same chief factors as an -injector for some Fitting class but which is not itself an -injector have been constructed by F.P. Lockett (in his PhD thesis) and T.R. Berger and John Cossey (in preparation). Graham A. Chambers (J. Algebra 16 (1970), 442–455) has shown that such a subgroup cannot exist in a group of p–length one for all primes p. The main result of this paper closes the small gap remaining: it includes Chambers' result and establishes also that such a subgroup cannot exist in a group of nilpotent length three.

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