Boolean lattices of n-multiply Ω-bicanonical Fitting classes

2002 ◽  
Vol 12 (5) ◽  
Author(s):  
O.V. Kamozina
Keyword(s):  
Boolean Lattice ◽  
Direct Decomposition ◽  
Fitting Class ◽  
Boolean Lattices ◽  
Fitting Classes ◽  
Direct Decompositions

AbstractWe describe the n-multiply Ω-bicanonical Fitting classes with Boolean lattice of Fitting subclasses. In particular, it is shown that in this case a Fitting class is directly decomposable with the use of the set of all atoms of its lattice. Here the notion of a direct decomposition plays the key role. Therefore we study direct decompositions separately and consider Ω-foliated Fitting classes with more general directions.

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 Isometries and direct decompositions of pseudo MV-algebras

2007 ◽  
Vol 57 (2) ◽  
Author(s):  
Milan Jasem
Keyword(s):  
Direct Decomposition ◽  
The Other ◽  
Other Hand ◽  
Mv Algebra ◽  
Direct Decompositions ◽  
Mv Algebras

AbstractIn the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra $$\mathcal{A}$$ = (A, ⊕, −, ∼, 0, 1) there exists an internal direct decomposition $$\mathcal{A} = \mathcal{B}^0 \times \mathcal{C}^0 $$ of $$\mathcal{A}$$ with $$\mathcal{C}^0 $$ commutative such that $$f(0) = 1_{C^0 } $$ and $$f(x) = x_{B^0 } \oplus (1_{C^0 } \odot (x_{C^0 } )^ - ) = x_{B^0 } \oplus (1_{C^0 } - x_{C^0 } )$$ for each x ∈ A.On the other hand, if $$\mathcal{A} = \mathcal{P}^0 \times \mathcal{Q}^0 $$ is an internal direct decomposition of a pseudo MV-algebra $$\mathcal{A}$$ = (A, ⊕, −, ∼, 0, 1) with $$\mathcal{Q}^0 $$ commutative, then the mapping g: A → A defined by $$g(x) = x_{P^0 } \oplus (1_{Q^0 } - x_{Q^0 } )$$ is an isometry in $$\mathcal{A}$$ and $$g(0) = 1_{Q^0 } $$ .

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.

scholarly journals Chains in generalized Boolean lattices

1976 ◽  
Vol 21 (2) ◽  
pp. 234-240
Author(s):  
Richard D. Byrd ◽  
Roberto A. Mena
Keyword(s):  
Distributive Lattice ◽  
Boolean Lattice ◽  
Boolean Lattices ◽  
A Chain

A chain C in a distributive lattice L is called strongly maximal in L if and only if for any homomorphism φ of L onto a distributive lattice K, the chain (Cφ)0 is maximal in K, where (Cφ)0 = Cφ if 0 ∉ K, and (Cφ)0 = Cφ ∪ {0}, otherwise. Gratzer (1971, Theorem 28) states that if B is a generalized Boolean lattice R-generated by L and C is a chain in L, then C R-generates B if and only if C is strongly maximal in L. In this note (Theorem 4.6), we prove the following assertion, which is not far removed from Gratzer's statement: let B be a generalized Boolean lattice R-generated by L and C be a chain in L. If 0 ∈ L, then C generates B if and only if C is strongly maximal in L. If 0 ∉ L, then C generates B if and only if C is strongly maximal in L and [C)L = L. In Section 5 (Example 5.1) a counterexample to Gratzer's statement is provided.

Subgroup closed Fitting classes

1978 ◽  
Vol 83 (2) ◽  
pp. 195-204 ◽  
Author(s):  
R. A. Bryce ◽  
John Cossey
Keyword(s):  
Saturated Formation ◽  
Fitting Class ◽  
Fitting Formation ◽  
Fitting Classes

In (1) we showed that a subgroup closed Fitting formation is a primitive saturated formation, and in (2) we showed that a subgroup closed and metanilpotent Fitting class is a formation. Whether or not a subgroup closed Fitting class is always a formation is a question that has plagued us ever since. The purpose of this paper is to prove

scholarly journals Fitting classes of certain metanilpotent groups

1994 ◽  
Vol 36 (2) ◽  
pp. 185-195 ◽  
Author(s):  
Hermann Heinenken
Keyword(s):  
Quotient Group ◽  
Nilpotent Groups ◽  
Fitting Class ◽  
Normal Subgroups ◽  
Supersoluble Groups ◽  
Nilpotent Residual ◽  
Saturated Formations ◽  
Open Question ◽  
Fitting Classes ◽  
Cyclic Commutator

There are two families of group classes that are of particular interest for clearing up the structure of finite soluble groups: Saturated formations and Fitting classes. In both cases there is a unique conjugacy class of subgroups which are maximal as members of the respective class combined with the property of being suitably mapped by homomorphisms (in the case of saturated formations) or intersecting suitably with normal subgroups (when considering Fitting classes). While it does not seem too difficult, however, to determine the smallest saturated formation containing a given group, the same problem regarding Fitting classes does not seem answered for the dihedral group of order 6. The object of this paper is to determine the smallest Fitting class containing one of the groups described explicitly later on; all of them are qp-groups with cyclic commutator quotient group and only one minimal normal subgroup which in addition coincides with the centre. Unlike the results of McCann [7], which give a determination “up to metanilpotent groups”, the description is complete in this case. Another family of Fitting classes generated by a metanilpotent group was considered and described completely by Hawkes (see [5, Theorem 5.5 p. 476]); it was shown later by Brison [1, Proposition 8.7, Corollary 8.8], that these classes are in fact generated by one finite group. The Fitting classes considered here are not contained in the Fitting class of all nilpotent groups but every proper Fitting subclass is. They have the following additional properties: all minimal normal subgroups are contained in the centre (this follows in fact from Gaschiitz [4, Theorem 10, p. 64]) and the nilpotent residual is nilpotent of class two (answering the open question on p. 482 of Hawkes [5]), while the quotient group modulo the Fitting subgroup may be nilpotent of any class. In particular no one of these classes consists of supersoluble groups only.

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 Ωζ-foliated Fitting Classes

2020 ◽  
Vol 20 (4) ◽  
pp. 424-433
Author(s):  
Olesia V. Kamozina ◽  
Keyword(s):  
Fitting Class ◽  
Class I ◽  
Simple Groups ◽  
Definition Of ◽  
Fitting Classes ◽  
Class F

All groups under consideration are assumed to be finite. For a nonempty subclass of Ω of the class of all simple groups I and the partition ζ = {ζi | i ∈ I}, where ζi is a nonempty subclass of the class I, I = ∪i∈I ζi and ζi ∩ ζj = ø for all i ≠ j, ΩζR-function f and ΩζFR-function φ are introduced. The domain of these functions is the set Ωζ ∪ {Ω′}, where Ωζ = { Ω ∩ ζi | Ω ∩ ζi ≠ ø }, Ω′ = I \ Ω. The scope of these function values is the set of Fitting classes and the set of nonempty Fitting formations, respectively. The functions f and φ are used to determine the Ωζ-foliated Fitting class F = ΩζR(f, φ) = (G : OΩ(G) ∈ f(Ω′) and G'φ(Ω ∩ ζi) ∈ f(Ω ∩ ζi) for all Ω ∩ ζi ∈ Ωζ(G)) with Ωζ-satellite f and Ωζ-direction φ. The paper gives examples of Ωζ-foliated Fitting classes. Two types of Ωζ-foliated Fitting classes are defined: Ωζ-free and Ωζ-canonical Fitting classes. Their directions are indicated by φ0 and φ1 respectively. It is shown that each non-empty non-identity Fitting class is a Ωζ-free Fitting class for some non-empty class Ω ⊆ I and any partition ζ. A series of properties of Ωζ-foliated Fitting classes is obtained. In particular, the definition of internal Ωζ-satellite is given and it is shown that every Ωζ-foliated Fitting class has an internal Ωζ-satellite. For Ω = I, the concept of a ζ-foliated Fitting class is introduced. The connection conditions between Ωζ-foliated and Ωζ-foliated Fitting classes are shown.

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