No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
On finite soluble products of N-connected groups
... More precisely, he proved that saturated formations (containing N ) are closed under the product of pairwise permutable and N -connected groups. This study was taken further in [3] in the soluble universe and for products of two N -connected groups, mainly in the framework of formation theory. As pointed out in this paper, although total permutability and N -connection are quite different properties, they are related in the sense that the first one is to supersolubility as the second one is to nilpotence. ...
... In [8] and [3] the behaviour of products of N -connected permutable subgroups with regard to formation theory was studied. In the sequel we will take this study further. ...
... The following lemma was proved in [3] for the soluble universe. ...
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.
... More precisely, he proved that saturated formations (containing N ) are closed under the product of pairwise permutable and N -connected groups. This study was taken further in [3] in the soluble universe and for products of two N -connected groups, mainly in the framework of formation theory. As pointed out in this paper, although total permutability and N -connection are quite different properties, they are related in the sense that the first one is to supersolubility as the second one is to nilpotence. ...
... and [3] the behaviour of products of N -connected permutable subgroups with regard to formation theory was studied. In the sequel we will take this study further. ...
... The following lemma was proved in [3] for the soluble universe. ...
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.
... More precisely, he proved that saturated formations (containing N ) are closed under the product of pairwise permutable and N -connected groups. This study was taken further in [3] in the soluble universe and for products of two N -connected groups, mainly in the framework of formation theory. As pointed out in this paper, although total permutability and N -connection are quite different properties, they are related in the sense that the first one is to supersolubility as the second one is to nilpotence. ...
... In [8] and [3] the behaviour of products of N -connected permutable subgroups with regard to formation theory was studied. In the sequel we will take this study further. ...
... The following lemma was proved in [3] for the soluble universe. ...
... In [1], Ballester-Bolinches and Pedraza-Aguilera proved that soluble N-connected products behave well with respect to saturated formations containing N. Following this idea, we study the behaviour of strongly cosubnormal subgroups in the finite (not necessarily soluble) universe with respect to formations. ...
... Consequently, A ∩ B Z ∞ (A) ∩ Z ∞ (B) which is contained in Z, by [5, Proposition 3.2]. On the other hand, [AZ/Z, BZ/Z] [A, B]Z/Z = 1, by Theorem 2, whence G/Z = AZ/Z × BZ/Z.(2) implies(1). Suppose that G/Z = AZ/Z × BZ/Z. ...
Two subgroups A and B of a group G are cosubnormal if A and B are subnormal in their join 〈A,B〉 and are strongly cosubnormal if every subgroup of A is cosubnormal with every subgroup of B. We find necessary and sufficient conditions for A and B to be strongly cosubnormal in 〈A,B〉 and, if Z is the hypercentre of G=〈A,B〉, we show that A and B are strongly cosubnormal if and only if G/Z is the direct product of AZ/Z and BZ/Z. We also show that projectors and residuals for certain formations can easily be constructed in such a group. Two subgroups A and B of a group G are N-connected if every cyclic subgroup of A is cosubnormal with every cyclic subgroup of B (N denotes the class of nilpotent groups). Though the concepts of strong cosubnormality and N-connectedness are clearly closely related, we give an example to show that they are not equivalent. We note, however, that if G is the product of the N-connected subgroups A and B, then A and B are strongly cosubnormal.
... 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. ...
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. ...
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). ...
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.
... noting the class of all finite nilpotent groups) have been thoroughly investigated in various papers (e.g. 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]). ...
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.
If k is a positive integer, a group G is said to have the -property if for each element g of G there exists a normal subgroup of finite index X(g) such that the subgroup is nilpotent of class at most k for all . Thus, -groups are precisely those groups with finite conjugacy classes ( FC -groups) and the aim of this paper is to extend properties of FC -groups to the case of groups with the -property for . The class of -groups contains the relevant subclass , consisting of all groups G for which to every element g there corresponds a normal subgroup of finite index Y(g) such that is nilpotent of class at most k , whenever U is a nilpotent subgroup of class at most k of Y(g) .
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.
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 .
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.
Subgroups A and B of a finite group are said to be 𝒩-connected if the subgroup generated by elements x and y is a nilpotent group, for every pair of elements x in A and y in B. The behaviour of finite pairwise permutable and 𝒩-connected products are studied with respect to certain classes of groups including those groups where all the subnormal subgroups permute with all the maximal subgroups, the so-called SM-groups, and also the class of soluble groups where all the subnormal subgroups permute with all the Carter subgroups, the so-called C-groups.
Two subgroups H and K of a group G are said to be totally permutable if every subgroup of H permutes with every subgroup of K. In this paper the behaviour of radicals and injectors associated to Fitting classes in a product of pairwise totally permutable finite groups is studied.
In this paper the structure of finite groups which are the product of two totally permutable subgroups is studied. In fact we can obtain the -residual, where is a formation, -projectors and -normalisers, where is a saturated formation, of the group from the corresponding subgroups of the factor subgroups.
Throughout the paper we consider only finite groups.
J. C. Beidleman and H. Smith [3] have proposed the following question: “If G is a group and H a subnormal subgroup of G containing Φ( G ), the Frattini subgroup of G , such that H /Φ( G )is supersoluble, is H necessarily supersoluble? “In this paper, we give not only an affirmative answer to this question but also we see that the above result still holds if supersoluble is replaced by any saturated formation containing the class of all nilpotent groups.
Saturated formations are closed under the product of subgroups which are connected by certain permutability properties.