Formations Of Finite Soluble Groups Research Articles

On the dual theory of a result of Bryce and Cossey

In the theory of formations of finite soluble groups, a well known result of Bryce and Cossey is: a local formation F is a Fitting class if and only if every value of the canonical formation function F of F is a Fitting class. In this paper, we give the dual theory of the result of Bryce and Cossey. We proved that an ω-local (in particular, local) Fitting class F is a formation if and only if every value of the canonical ω-local (in particular, local) Hartley function F of F is a formation.

On a class of finite soluble groups

Abstract The aim of this paper is to study the class of finite groups in which every subgroup is self-normalising in its subnormal closure. It is proved that this class is a subgroup-closed formation of finite soluble groups which is not closed under taking Frattini extensions and whose members can be characterised by means of their Carter subgroups. This leads to new characterisations of finite soluble T-, PT- and PST-groups. Finite groups whose p-subgroups, p a prime, are self-normalising in their subnormal closure are also characterised.

ON LOCALLY DEFINED FORMATIONS OF SOLUBLE LIE AND LEIBNIZ ALGEBRAS

AbstractIt is well known that all saturated formations of finite soluble groups are locally defined and, except for the trivial formation, have many different local definitions. I show that for Lie and Leibniz algebras over a field of characteristic 0, the formations of all nilpotent algebras and of all soluble algebras are the only locally defined formations and the latter has many local definitions. Over a field of nonzero characteristic, a saturated formation of soluble Lie algebras has at most one local definition, but a locally defined saturated formation of soluble Leibniz algebras other than that of nilpotent algebras has more than one local definition.

The G-strong containment for locally defined formations of finite soluble groups

Let F and H be two saturated formations of finite soluble groups, with canonical definition F and H, respectively, and Char ( F ) ⊆ Char ( H ) . We say that F is G -strong containment in H , when for all p ∈ Char ( F ) and for all H ∈ H the H ( p ) -Residual of H is in a F -projector E of H containment. In this case we write F ≪ G H . In this work we wants to show a characterization of the saturated formation H , which contain G -strongly the formations S π , NX , N k and U ¨ , respectively.

Nilpotent-like fitting formations of finite soluble groups

Dedicated to Professor K. Doerk on his 60th Birthday.In this paper the subnormal subgroup closed saturated formations of finite soluble groups containing nilpotent groups are fully characterised by means of extensions of well-known properties enjoyed by the formation of all nilpotent groups.

A Gaschütz–Lubeseder Type Theorem in a Class of Locally Finite Groups

The aim of this paper is to present a Gaschutz–Lubeseder type theorem in the class cL of all radical locally finite groups satisfying min−p for all primes p. Notice that these groups are countable and co-Hopfian by [1, (5.4.8)]. In retrospect, the theory of saturated formations of finite soluble groups began with the results of Gaschutz [3] in 1963. He introduced the concept of “covering subgroup” as a generalization of Sylow and Hall subgroups. These covering subgroups have many of the properties of Sylow and Hall subgroups other than the arithmetic ones. The main idea of Gaschutz’s work was concerned with group theoretical classes having the same properties. He defined a formation F to be a ŒQ;R0-closed class of finite groups and he called F saturated if a group G ∈ F whenever the Frattini fac-

On a theorem of Bryce and Cossey

In this paper we characterise the subgroup–closed Fitting formations of finite groups which are saturated. This is an extension of the Bryce and Cossey result proving the saturation of all subgroup-closed Fitting formations of finite soluble groups.

A question in the theory of saturated formations of finite soluble groups

This paper examines the following question. If\(\mathcal{H}\) and\(\mathcal{F}\) are saturated formations then\(\mathcal{H}_\mathcal{F} \) is defined to be the class of all soluble groups whose\(\mathcal{H} - normalizers\) belong to\(\mathcal{F}\). In general\(\mathcal{H}_\mathcal{F} \) is a formation, but need not be a saturated formation. Here the smallest saturated formation containing\(\mathcal{H}_\mathcal{F} \) is studied.

Local definitions of local homomorphs and formations of finite groups

It is well known that every local formation of finite soluble groups possesses three distinguished local definitions consisting of finite soluble groups: the minimal one, the full and integrated one, and the maximal one. As far as the first and the second of these are concerned, this statement remains true in the context of arbitrary finite groups. Doerk, Šemetkov, and Schmid have posed the problem of whether every local formation of finite groups has a distinguished (that is, unique) maximal local definition. In this paper a description of local formations with a unique maximal local definition is given, from which counter-examples emerge. Furthermore, a criterion for a formation function to be a local definition of a given local formation is obtained.

On generalized covering subgroups and a characterisation of ?pronormal?

Introduction. The context of this note is the theory of Schunck classes and formations of finite soluble groups. In a 1972 manuscript Fischer [4] generalized the concept of an ~-covering subgroup of a group G to a (P, ~)-covering subgroup, where P is some pronormal subgroup of G, and proved universal existence (for P satisfying a stronger embedding property) in case the class ~ is a saturated formation. The fact tha t the Schunck classes are the classes ~ with the property that every group has an ~-projector [9, 4.3, 4.4; 6] (which coincides with an ~-covering subgroup in the soluble universe | [6, II.15]) raises the question whether it is possible to determine the whole range of universal existence of (P, ~)-covering subgroups. The aim of this paper is to show that such classes are exactly the Schunck classes ~ of form ~ ~ E~ ~ for some formation ~. In a short second part we wish to note that the embedding property "pronormal" can be characterized by the Fratt ini argument.

Splitting theorems in abelian-by-hypercyclic groups

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.

On saturated formations which are special relative to the strong covering-avoidance property

Let F \mathfrak {F} be a saturated formation of finite soluble groups. Let G G be a finite soluble group and F F an F \mathfrak {F} -projector of G G . Then F F is said to satisfy the strong covering-avoidance property if (i) F F either covers or avoids each chief factor of G G , anaaa F ∩ L / F ∩ K F \cap L/F \cap K is a chief factor of F F whenever L / K L/K is a chief factor of G G coverzd by F F Let C F {\mathcal {C}_\mathfrak {F}} denote the class of all finite soluble G G in which the F \mathfrak {F} -projectors satisfy the strong covering-avoidance property. C F {\mathcal {C}_\mathfrak {F}} is a formation. Let Y F {\mathcal {Y}_\mathfrak {F}} be the class of groups G G in which an F \mathfrak {F} -normalizer is also an F \mathfrak {F} -projector. Y F {\mathcal {Y}_\mathfrak {F}} aa is a formation studied by Klaus Doerk. Note that Y F ⊆ C F {\mathcal {Y}_\mathfrak {F}} \subseteq {\mathcal {C}_\mathfrak {F}} . F \mathfrak {F} is said to be C \mathcal {C} -special if C F = Y F {\mathcal {C}_\mathfrak {F}} = {\mathcal {Y}_\mathfrak {F}} . he puraaaaaaaaaaas note is to sdy C \mathcal {C} -special formations. Two characterizations of C \mathcal {C} -special formations are given. Let i i be a positive integer and let N ( i ) {\mathfrak {N}^{(i)}} denote the class of finite soluble groups G G whose Fitting length is at most i i . Then N ( i ) {\mathfrak {N}^{(i)}} is C \mathcal {C} -special. Finally, the formation C F {\mathcal {C}_\mathfrak {F}} aaaaaturated if and only if F \mathfrak {F} is the class of all finite soluble groups.

On Huppert’s condition $B$

Let $\mathcal {F}$ be a saturated formation of finite soluble groups. $\mathcal {F}$ is said to satisfy condition $\text {B}$ if and only if (a) $\mathcal {F}$ is subgroup-closed, and (b) $G \in \mathcal {F}$ and $N$ a minimal normal subgroup of $G$ implies $\operatorname {Aut} (N) \in \mathfrak {F}$. The purpose of this note is to characterize those saturated formations of finite soluble groups which satisfy condition ${\mathbf {B}}$.

An analogue of lubeseder's theorem for formations of finite rings

For formations of finite soluble groups, the properties of Frattini closure and local defineability are known to be equivalent (see [2]). The investigations of Barnes and Gastineau-Hills [1] on the other hand reveal that although every Frattini closed formation of finite-dimensional soluble Lie algebras over an algebraically closed field of zero characteristic is local, without the algebraic closure condition the relationship between the two properties breaks down even for supersoluble Lie algebras. We are concerned here with the analogous problem for rings. The main results are;

Fitting formations of finite soluble groups

Fitting formations of finite soluble groups

On the definition of saturated formations of groups

We exhibit a closure operation which serves to define saturated formations of finite soluble groups.

Formations and a class of locally soluble groups

In (3) W. Gaschütz introduced the concept of a formation of finite soluble groups, generalizing the results of R. W. Carter on the existence and conjugacy of nilpotent self-normalizing subgroups in finite soluble groups (2). Carter's results have already been extended to two classes of infinite groups in (9), (10), and the object of the present work is to extend the results of Gaschütz to a class of infinite groups, thereby generalizing (9).