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Saturated formations and products of connected subgroups
Abstract
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.
... The structure and properties of N -connected products, for the class N of finite nilpotent groups, are well known (cf. [7][8][9]); for instance, G = AB is an N -connected product of A and B if and only if G modulo its hypercenter is a direct product of the images of A and B. Apart from the above-mentioned results regarding S-connection, corresponding studies for the classes N 2 and N A of metanilpotent groups, and groups with nilpotent derived subgroup, respectively, have been carried out in [10,11]; in [12] connected products for the class S π S ρ of finite soluble groups that are extensions of a normal π-subgroup by a ρ-subgroup, for arbitrary sets of primes π and ρ, are studied. The class S π S ρ appears in that reference as the relevant case of a large family of formations, named nilpotent-like Fitting formations, which comprise a variety of classes of groups, such as the class of π-closed soluble groups, or groups with Sylow towers with respect to total orderings of the primes. ...
... In the present paper, as an application of Theorem 1, we show that the main results in [10][11][12], proved for soluble groups, remain valid for arbitrary finite groups. In particular, we characterize connected products for some relevant classes of groups (see Theorem 4). ...
... We gather next our main results. The first one extends to the universe of finite groups results for soluble groups in [11] (Theorem 3), [10] (Theorem 1, Proposition 1) and [12] (Theorem 1). 3. Let π, ρ be arbitrary sets of primes. The following are equivalent: ...
For a non-empty class of groups L, a finite group G=AB is said to be an L-connected product of the subgroups A and B if ⟨a,b⟩∈L for all a∈A and b∈B. In a previous paper, we prove that, for such a product, when L=S is the class of finite soluble groups, then [A,B] is soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite groups previous research about finite groups in the soluble universe. In particular, we characterize connected products for relevant classes of groups, among others, the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of relevant subgroups of finite groups.
... Structure and properties of N -connected products, for the class N of finite nilpotent groups, are well known (cf. [1,14,2]); for instance, G = AB is an N -connected product of A and B if and only if G modulo its hypercenter is a direct product of the images of A and B. Apart from the above-mentioned results regarding S-connection, corresponding studies for the classes N 2 and N A of metanilpotent groups, and groups with nilpotent derived subgroup, respectively, have been carried out in [8,9]; in [10] connected products for the class S π S ρ of finite soluble groups that are extensions of a normal π-subgroup by a ρ-subgroup, for arbitrary sets of primes π and ρ, are studied. The class S π S ρ appears in that reference as the relevant case of a large family of formations, named nilpotent-like Fitting formations, which comprise a variety of classes of groups, such as the class of π-closed soluble groups, or groups with Sylow towers with respect to total orderings of the primes. ...
... In the present paper, as an application of Theorem 1.1, we show that main results in [8,9,10], proved for soluble groups, remain valid for arbitrary finite groups. In particular, we characterize connected products for some relevant classes of groups (see Theorem 1.6). ...
... Remark 1. 10. Easy examples show that the bound for the π-length of G in Corollary 1.9 is sharp. ...
For a non-empty class of groups , a finite group is said to be an -connected product of the subgroups A and B if for all and . In a previous paper, we prove that for such a product, when is the class of finite soluble groups, then [A,B] is soluble. This generalizes the theorem of Thompson which states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper our result is applied to extend to finite groups previous research in the soluble universe. In particular, we characterize connected products for relevant classes of groups; among others the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Also we give local descriptions of relevant subgroups of finite groups.
... For the special case when G = AB = A = B this means of course that a, b ∈ L for all a, b ∈ G, and the study of products of L-connected subgroups provides a more general setting for local-global questions related to two-generated subgroups. We refer to [8,28,9] for previous studies for the class L = N of finite nilpotent groups, and to [18,19,20,21] for L being the class of finite metanilpotent groups and other relevant classes of groups. For the class L = S of finite soluble groups, A. Carocca in [12] proved the solubility of a product of S-connected soluble subgroups, which provides a first extension of the above-mentioned theorem of Thompson for products of groups (see Corollary 2). ...
... In particular, Corollary 2 generalizes Carocca's result via the soluble radical in a product of S-connected subgroups. In a forthcoming paper [17], our theorem is applied to extend main results known for finite soluble groups in [18,19,20] to the universe of all finite groups. ...
A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, n-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups with subgroups such that is soluble for all and . In this case, the group G is said to be an -connected product of the subgroups A and B for the class of all finite soluble groups. Our main theorem states that is -connected if and only if [A,B] is soluble. In the course of the proof we derive a result of own interest about independent primes regarding the soluble graph of almost simple groups.
... Finally, a study in [12] deals with the influence of 2-generator subgroups on the structure of groups and factorized groups. Here nilpotent-like Fitting formations of soluble groups are characterized like certain saturated formations F of soluble groups which are closed under taking products of F-connected F-subgroups. ...
... For instance, given any sets π and ρ of primes, the class E π E ρ of all groups that are extensions of π-groups by ρ-groups is a nilpotent-like Fitting formation as well as the class of all groups with a Sylow tower for any given total ordering of the primes. Many more examples are described in [12]. Let G be a group and G p a Sylow p-subgroup of G for any prime p ∈ π(G), the set of prime divisors of the order of G. Then the automiser A p (G) is the group N G (G p )/G p C G (G p ), up to conjugacy of Sylow subgroups. ...
We report on a stream of research in relation with Sylow normalizers, i.e. normalizers of Sylow subgroups, of finite groups and group's classes.
A remarkable result of Thompson states that a finite group is soluble if and only if all its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, n-generated) subgroups. We contribute an extension of Thompson’s theorem from the perspective of factorized groups. More precisely, we study finite groups with subgroups such that is soluble for all and . In this case, the group G is said to be an -connected product of the subgroups A and B for the class of all finite soluble groups. Our Main Theorem states that is -connected if and only if [A, B] is soluble. In the course of the proof, we derive a result about independent primes regarding the soluble graph of almost simple groups that might be interesting in its own right.
In this paper we study groups G generated by two subgroups A and B such that 〈. a, b〉 is nilpotent of class at most 2 for all a∈. A and b∈. B. A detailed description of the structure of such groups is obtained, generalizing the classical result of Hopkins and Levi on 2-Engel groups.
Let F be a non-empty subgroup-closed saturated formation, it is studied that the structure of a finite group G under the assumption that every subgroup of prime order p if p≠ 2 and every cyclic subgroup of order 2 or 4 if p=2 are contained in the F-hypercenter of the normalizer N
G
(P) of a Sylow p-subgroup P of G. Our results improve and extend some conclusions in the previous references.
Two subgroups H and K of a finite group G are said to be N-connected if the subgroup generated by x and y is a nilpotent group, for every pair of elements x in H and y in K. This paper is devoted to the study of pairwise N-connected and permutable products of finitely many groups, in the framework of formation and Fitting class theory.
In this article we show that a finite soluble group possesses nilpotent Hall subgroups for well-defined sets of primes if and only if its Sylow normalizers satisfy the same property. In fact, this property of groups provides a characterization of the subgroup- closed saturated formations, whose elements are characterized by the Sylow normalizers belonging to the class, in the universe of all finite soluble groups.
Let G be a finite group and be a Sylow p-subgroup of G for a prime p in , the set of all prime divisors of the order of G. The automiser is defined to be the group . We define the Sylow graph of the group G, with set of vertices , as follows: Two vertices form an edge of if either or . The following result is obtained: Theorem: Let G be a finite almost simple group. Then the graph is connected and has diameter at most 5. We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.
In this second paper, the bulk of the work is devoted to characterizing E2(3) and S4(3). These two groups are “almost” AT-groups and it is relevant to treat them separately. The actual characterizations (Theorems 8.1 and 9.1) are very technical but the hypotheses deal with the structure and embedding in a simple group of certain {2, 3}-subgroups.
The main result in the paper states the following: For a finite
group G=AB, which is the product of the soluble subgroups A
and B, if is a metanilpotent group for all
and , then the factor groups are nilpotent, F(G) denoting the Fitting subgroup of
G. A particular generalization of this result and some
consequences are also obtained. For instance, such a group G is
proved to be soluble of nilpotent length at most l+1, assuming
that the factors A and B have nilpotent length at most l. Also
for any finite soluble group G and , an element
is contained in the preimage of the hypercenter of ,
where denotes the ()th term of the Fitting series
of G, if and only if the subgroups have
nilpotent length at most k for all .
In this paper, the simple N-groups are classified for which e ≧ 3 and 2 ∈ π4. This latter condition means that a Sylow 2-subgroup contains a normal elementary abelian subgroup of order 8 and does not normalize any nonidentity odd order subgroup.
For a non-empty class of groups ℱ, two subgroups A and B of a finite group G are said to be ℱ-connected if 〈a, b〉 ∈ ℱ for all a ∈ A and b ∈ B. This paper is a study of ℱ-connection for saturated formations ℱ ⊆ (where denotes the class of all finite groups with nilpotent commutator subgroup). The class of all finite supersoluble groups constitutes an example of such a saturated formation. It is shown for example that in a finite soluble group G = AB the subgroups A and B are -connected if and only if [A, B] ⩽ F(G), where F(G) denotes the Fitting subgroup of G. Also ℱ-connected finite soluble products for any saturated formation ℱ with ℱ ⊆ are characterized.
Saturated formations are closed under the product of subgroups which are connected by certain permutability properties.
Saturated formations which contain the class of supersoluble groups are closed under the product of two subgroups if every subgroup of one factor is permutable with every subgroup of the other.
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
Using classification theorems of
simple groups, we give a proof of a conjecture on factorized finite groups which is an extension of a
well known theorem due to P. Hall.
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