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Products of finite connected subgroups
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Abstract
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.
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We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.
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.
We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x in G the subgroup of G generated by x and y is solvable. We present analogues of this result for finite dimensional Lie algebras and some classes of infinite groups. We also consider a similar problem for pairs of elements.
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 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.
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 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.
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.
Saturated formations are closed under the product of subgroups which are connected by certain permutability properties.
Suppose G = AB where G is a finite group and A and B are nilpotent subgroups. It is proved that the derived length of G modulo its Frattini subgroup is at most the sum of the classes of A and B. An upper bound for the derived length of G in terms of the derived lengths of A and B also is obtained.
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.
Finite groups in which every two elements generate a soluble group
P. Flavell, Finite groups in which every two elements generate a
soluble group, Inventiones Math. 121 (1995), 279-285.
A characterization of F 2 (G)
P. Flavell, A characterization of F 2 (G), J. Algebra 255 (2002), 271-287.
Sand 13, 72076 Tübingen, Germany E-mail: peter.hauck@uni-tuebingen.de L
- M P Gállego Departamento De Matemáticas
M. P. GÁLLEGO
Departamento de Matemáticas, Universidad de Zaragoza,
Edificio Matemáticas, Ciudad Universitaria, 50009 Zaragoza, Spain
E-mail: pgallego@unizar.es
P. HAUCK
Fachbereich Informatik, Universität Tübingen,
Sand 13, 72076 Tübingen, Germany
E-mail: peter.hauck@uni-tuebingen.de
L. S. KAZARIN
Department of Mathematics, Yaroslavl P. Demidov State University
Sovetskaya Str 14, 150014 Yaroslavl, Russia
E-mail: Kazarin@uniyar.ac.ru
A. MARTÍNEZ-PASTOR
Instituto Universitario de Matemática Pura y Aplicada IUMPA
Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain
E-mail: anamarti@mat.upv.es
and M. D. PÉREZ-RAMOS
Departament de Matemàtiques, Universitat de València,
C/ Doctor Moliner 50, 46100 Burjassot (València), Spain
E-mail: Dolores.Perez@uv.es