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Thompson-like characterization of solubility for products of finite groups
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Abstract
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.
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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 present Ramanujan's proof of Bertrand's postulate and in the process, eliminate his use of Stirling's formula. The revised proof is elegant and elementary so as to be accessible to a wider audience.
Many group theorists all over the world have been trying in the last twenty-five years to extend and adapt the magnificent methods of the Theory of Finite Soluble Groups to the more ambitious universe of all finite groups. This is a natural progression after the classification of finite simple groups but the achievements in this area are scattered in various papers. Our objectives in this book were to gather, order and examine all this material, including the latest advances made, give a new approach to some classic topics, shed light on some fundamental facts that still remain unpublished and present some new subjects of research in the theory of classes of finite, not necessarily solvable, groups.
In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x∈C and y∈D for which 〈x, y〉 is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof
of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y∈G with |x|=a and |y|=b, the subgroup 〈x, y〉 is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any
family of finite groups closed under forming subgroups, quotients and extensions.
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.
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.
Supersoluble immersion of a normal subgroup K of a finite group G shall be defined by the following property:
If σ is a homomorphism of G, and if the minimal normal subgroup J of G σ is part of K σ then J is cyclic (of order a prime).
Our principal aim in the present investigation is the proof of the equivalence of the following three properties of the normal subgroup K of the finite group G:
(i) K is supersolubly immersed in G.
(ii) K/ϕK is supersolubly immersed in G/ϕK.
(iii) If θ is the group of automorphisms induced in the p-subgroup U of K by elements in the normalizer of U in G, then θ' θ p-1 is a p-subgroup of θ .
Though most of our discussion is concerned with the proof of this theorem, some of our concepts and results are of independent interest.
The next three chapters are primarily devoted to studying primitive groups. Primitive groups play an important role as building blocks, particularly in the study of finite permutation groups. Frequently, we can carry out a series of reductions: from the general case to the transitive case by examining the action of the group on its orbits and its point stabilizers, and then from the transitive imprimitive case to the primitive case by studying the action of the group on sets of blocks and the block stabilizers. Eventually, at least for finite permutation groups, this reduces the original question to one about primitive groups. Of course, this is rarely the whole problem; generally we must then retrace the process, fitting the information back together as we reconstruct the original group, and often this is very complicated. Still, the crux of many problems in finite permutation groups lies in the study of the primitive case.
This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of the geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.
The maximal independent sets of the soluble graph of a finite simple group G are studied and their independence number is determined. In particular, it is shown that this graph in many cases has an independent set with three vertices.
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.
With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the maximal subgroups of the finite classical groups and present research into these groups as well as proving many new results. In particular, the authors develop a unified treatment of the theory of the 'geometric subgroups' of the classical groups, introduced by Aschbacher, and they answer the questions of maximality and conjugacy and obtain the precise shapes of these groups. Both authors are experts in the field and the book will be of considerable value not only to group theorists, but also to combinatorialists and geometers interested in these techniques and results. Graduate students will find it a very readable introduction to the topic and it will bring them to the very forefront of research in group theory.
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.
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.
Some criteria of the non-simplicity of a finite group by graph theoretical terms are derived. This is then used to establish conditions under which a finite group is soluble.
Guralnick, Kunyavskii, Plotkin and Shalev have shown that the solvable radical of a finite group G can be characterized as the set of all x is an element of G such that < x, y > is solvable for all y is an element of G. We prove two generalizations of this result. Firstly, it is enough to check the solvability of < x, y > for every p-element y is an element of G for every odd prime p. Secondly, if x has odd order, then it is enough to check the solvability of < x, y > for every 2-element y is an element of G.
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.
Each finite simple group other thanU
3(3) can be generated by three of its involutions. In fact, each such group is generated by two elements, of which one is strongly real and the other is an involution.
We prove that an element g of prime order >3 belongs to the solvable radical R(G) of a finite (or, more generally, a linear) group if and only if for every x∈G the subgroup generated by g, xgx−1 is solvable. This theorem implies that a finite (or a linear) group G is solvable if and only if in each conjugacy class of G every two elements generate a solvable subgroup.
We prove that an element g of prime order q > 3 belongs to the solvable radical R(G) of a finite group if and only if for every x is an element of G the subgroup generated by g and xgx(-1) is solvable. This theorem implies that a finite group G is solvable if and only if ill each conjugacy class of G every two elements generate a solvable subgroup. To cite this article: N. Gordeev et al., C R. Acad. Sci. Paris, Ser. I 347 (2009). (C) 2009 Published by Elsevier Masson SAS oil behalf of Academie des sciences.
L k }, it follows that N ≤ A which contra-References [1] S. Abe, N. Iiyori, A generalization of prime graphs of finite groups
As A acts transitively on {L 1,..., L k }, it follows that N ≤ A which contra-References
[1] S. Abe, N. Iiyori, A generalization of prime graphs of finite groups.
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3 (1892) 265-284.
Germany e-mail: peter.hauck@uni-tuebingen
- P Hauck Fachbereich
- Informatik
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