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Locally defined Fitting classes
Abstract
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
... In 1967, Fischer, Gaschütz and Hartley found an important generalization of Sylow theorem and Hall theorem in the theory of Fitting classes (see [5]). Based on this, Hartley [9] and D'Arcy [3] established the theory of local Fitting classes, as follows. Let f be a function P → {Fitting classes}, which was later called a Hartley function or, for brevity, an H -function (see, for example, [15]), and ...
... Now assume that an F-injector V of G covers some p-chief factor H/K of G. Obviously, a Sylow p-subgroup P of V covers H/K . If p / ∈ π , then N p F (see [3,Lemma 2.3] or [4, Theorem IX.1.9]) and so the Sylow p-subgroup P of V is trivial. ...
... is not a Fitting class. Let Y = SL(2,3) be the special linear group of degree 2 over the field GF(3). Then the order of Y is 24 and Y contains the unique minimal normal groupD = Z (Y ) of order 2. Let H = Y 1 × Y 2 , where Y i Y for i = 1, 2 and D = {(d, d) | d ∈ D}.Then D is a minimal normal subgroup of H with order 2.By[4, B.9.16], there exists a faithful irreducible representation Φ of H/D over GF(3). ...
In this paper, we establish the theory of F-centrality of chief factors and F-hypercentre for Fitting classes. Based on this, we prove that every F-injector of G covers each F-central chief factor of G and avoids each F-eccentric chief factor of G if the Fitting class F has an integrated and invariable H-function. As an application, we also give a new method to define local Fitting classes, that is, we use local functions but not H-functions to define local Fitting classes.
... Recall that a formation F is called local if there exists such a function f : P → {formations of groups} that a group G ∈ F if and only if G/O p ′ , p (G) ∈ f (p) for every prime p dividing the order |G| of G and in this case they write F = LF (f ). Analogously, a Fitting class F is called local [1,2] if there exists such a function f : P → {Fitting classes} that G ∈ F if and only if O p, p ′ (G) ∈ f (p) for every prime p dividing |G| where O p, p ′ (G) = (G G p ′ ) Np (i.e. O p, p ′ (G) is the N presidual of the G p ′ -residual G G p ′ of G). ...
... 2. A Fitting class H of soluble groups is said to be locally defined if there exists a family {h(p) | p ∈ Π} of Fitting classes h(p), for every p in a set of primes Π , such that H = S Π ∩ p∈Π h(p)S p S p ; [10]. We point out in addition that nilpotent-like Fitting formations are exactly those Fitting classes locally defined by a family {S σ (p) | p ∈ τ } where σ (p) is a set of primes containing p, for every p in a set of primes τ . ...
For a non-empty class of groups C, two subgroups A and B of a group G are said to be C-connected if 〈a,b〉∈C for all a∈A and b∈B. Given two sets π and ρ of primes, SπSρ denotes the class of all finite soluble groups that are extensions of a normal π-subgroup by a ρ-group.It is shown that in a finite group G=AB, with A and B soluble subgroups, then A and B are SπSρ-connected if and only if Oρ(B) centralizes AOπ(G)/Oπ(G), Oρ(A) centralizes BOπ(G)/Oπ(G) and G∈Sπ∪ρ. Moreover, if in this situation A and B are in SπSρ, then G is in SπSρ.This result is then extended to a large family of saturated formations F, the so-called nilpotent-like Fitting formations of soluble groups, and to finite groups that are products of arbitrarily many pairwise permutable F-connected F-subgroups.
By a Fitting set of a group G one means a nonempty set of subgroups
of a finite group G which is closed under taking normal subgroups, their products, and conjugations of subgroups. In the present paper, the existence and conjugacy of
-injectors of a partially π-solvable group G is proved and the structure of -injectors is described for the case in which is a Hartley set of G.
We describe some methods for constructing Fischer classes of finite groups by means of the operators defined by given properties of Hall π-subgroups. It is in particular proved that, for a Fischer class F and a set of primes π, the class of all finite π-soluble CπF-groups, i.e., of all groups whose Hall π-subgroups belong to F, is a Fischer class.
A Fitting class FF is called dominant in the class of all finite soluble groups SS if F⊆SF⊆S and for every group G∈SG∈S any two FF-maximal subgroups of G containing the FF-radical GFGF of G are conjugate in G. In this paper a characterization of dominant local Fitting classes in the class of all finite soluble groups is established.
In this article, we give the description of the -injectors of a finite soluble group, for a Hartley class .
- W Fischer
- B Gaschiitz
- Hartley
Fischer, W. Gaschiitz, and B. Hartley (1967), 'Injektoren endlicher auflosbarer Gruppen',
Math. Z. 102, 337-339.