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On 2-generated subgroups and products of groups
Abstract
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.
... 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 1.15. As application of Theorem 1.6, the hypothesis of solubility can be also omitted in Corollary 4 and Propositions 3, 4 of [9], especially in relation with saturated formations F ⊆ N A, such as the class of supersoluble groups. Also an extension for finite groups of Corollary 1 of [10], in relation with the above-mentioned nilpotent-like Fitting formations, can be stated. ...
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.
... Public Health 2019, 16, 1570 2 of 11 Improved treatment of HF has resulted in decreased mortality and hospitalization rates [8], and thus, in addition to prolonging survival, increasing self-perceived health-related quality of life (HRQoL) has become a major goal of HF treatment [9]. Although there is no consensus on the precise definition, HRQoL captures a person's self-perceived impact of a medical condition, its symptoms, and its treatment [10]. It is highly individual and can mean different things to each patient, depending on demographic, psychological, socioeconomic, and other characteristics, in addition to interpretation based on the patient's own expectations, hopes, and ambitions [11]. ...
(1) Background: the main objective of this study was to investigate information needs concerning sexual activity and experienced sexual problems in heart failure (HF) patients and, in addition, to examine the association between these sexual problems and health-related quality of life (HRQoL); (2) Methods: in this cross-sectional study, three self-administered questionnaires were distributed to 77 stable ambulatory HF patients to acquire data on HRQoL, sexual problems, and need for counselling; (3) Results: More than half (56.7%) of HF patients experienced a marked decrease or total cessation of sexual activity due to their illness. Additionally, more than one-third perceived a marked decrease or total absence of sexual pleasure (42.5%), interest (32.9%), and constant problems or being unable to perform sexual activity (37.3%). Furthermore, 43.1% of patients experienced an important overall need for counselling concerning sexual activity, with information on relationships (69.2%), symptoms (58.5%), and relaxation (49.2%) being the most desired topics. Multiple linear regression analysis revealed that sexual problems were independently associated with HRQoL, with more sexual problems (t = 3.19, p < 0.01) being related to poor HRQoL; (4) Conclusion: by investigating the experienced problems and counselling needs of HF patients, an alignment between current practice and HF patients’ expectations and needs might be obtained.
... Ballester-Bolinches and Pedraza-Aguilera [2] and Hauck, Martínez-Pastor and Pérez-Ramos [13]); for instance, G = A B 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. For N 2 , the class of metanilpotent groups, and for saturated formations F ⊆ N A, the class of nilpotent-by-abelian groups, there are also very satisfactory characterizations of finite (soluble) groups which are products of F -connected subgroups (cf. Gállego, Hauck, Pérez-Ramos [15,14]). The class of all finite supersoluble groups is a relevant example of this latter family of saturated formations. ...
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.
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.
Two subgroups X and Y of a group G are said to be conditionally permutable in G if X permutes with Yg for some element g∈G, i.e., XYg is a subgroup of G. Using this permutability property new criteria for the product of finite supersoluble groups to be supersoluble are obtained and previous results are recovered. Also the behaviour of the supersoluble residual in products of finite groups is studied.
Two subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) in G if X permutes with Yg for some g ∊ 〈X, Y〉, for all X ≤ A and Y ≤ B. We study the belonging of a finite product of tcc-permutable subgroups to a saturated formation of soluble groups containing all finite supersoluble groups.
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.
This paper provides a supersoluble analogue of Engel theory. We show that the theory works for finite groups and list some further results pertaining to finitary skew linear groups.
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.
Saturated formations are closed under the product of subgroups which are connected by certain permutability properties.
Two subgroups H and K of a finite group G are said to
be -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 -connected and permutable products of
finitely many groups, in the framework of formation and
Fitting class theory.
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.