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Finite groups in which every two elements generate a soluble subgroup
... A similar result for two-generated subgroups with respect to the solvability property is provided by a remarkable theorem of J. Thompson [Th], [Fl1]: ...
... The result of Theorem 5.4 was conjectured by P. Flavell in 1997 [Fl4] (see also [Fl5]). After obtaining in [Fl1] a short proof of Thompson's theorem, he raised a natural question what happens if we keep one of the generators fixed and conjectured that we arrive at the solvable radical. ...
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.
... A similar result for two-generated subgroups with respect to the solvability property is provided by a remarkable theorem of J. Thompson [Th], [Fl1]: ...
... The result of Theorem 5.4 was conjectured by P. Flavell in 1997 [Fl4] (see also [Fl5]). After obtaining in [Fl1] a short proof of Thompson's theorem, he raised a natural question what happens if we keep one of the generators fixed and conjectured that we arrive at the solvable radical. ...
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.
... Theorem 2.6 [28], [8] Let G be a finite group in which every two elements generate a solvable subgroup. Then G is solvable. ...
... In [5] it was proved that the identity v 3 ≡ v n can hold in certain finite simple groups such as PSL (2,4), PSL (2,8), etc. Let us also mention a pioneer result of N. Gupta [10]: any finite group satisfying the identity v 1 ≡ v n is abelian. ...
We study two-variable Engel-like relations and identities characterizing finite-dimensional solvable Lie algebras and, conjecturally, finite solvable groups and introduce some invariants of finite groups associated with such relations. Bibliography: 29 titles.
... Theorem 2.6 [28], [8] Let G be a finite group in which every two elements generate a solvable subgroup. Then G is solvable. ...
... In [5] it was proved that the identity v 3 ≡ v n can hold in certain finite simple groups such as PSL (2,4), PSL (2,8), etc. Let us also mention a pioneer result of N. Gupta [10]: any finite group satisfying the identity v 1 ≡ v n is abelian. ...
We study two-variable Engel-like relations and identities characterizing finite dimensional solvable Lie algebras and, conjecturally, finite solvable groups, and introduce some invariants of finite groups associated to such relations. # This research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities --- Center of Excellence Program. + Partially supported by the Ministry of Absorption (Israel) and the Minerva Foundation through the Emmy Noether Research Institute of Mathematics. 1 Motivation Our primary interest in problems discussed in the present paper came from a paper by Segev [25] where the Margulis--Platonov conjecture had been related to some properties of the commuting graph of a finite simple group. (Given a finite group F , its commuting graph #(F ) has vertices indexed by the elements of F di#erent from 1; x, y # F are joined by an edge if and only if they commute.) In a more recent paper [26] the expected properties...
... The question "is a given finite group solvable?" can be answered by examining all 2-generated subgroups of the group [7]. A more complicated and celebrated result is Thompson's classification of N -groups, finite groups all of whose subgroups are either solvable or Fitting-free [16]. ...
We define and investigate the property of being “exponent-critical” for a finite group. A finite group is said to be exponent-critical if its exponent is not the least com-
mon multiple of the exponents of its proper non-abelian subgroups. We explore properties of exponent-critical groups and give a characterization of such groups. This characterization generalizes a classical result of Miller and Moreno on minimal non-abelian groups;
interesting families of p-groups appear.
... Baer [4] proved that G is a supersolvable group if and only if all the subgroups of G generated by two elements are supersolvable groups and that G is an F nn -group if and only if all the subgroups of G generated by three elements are F nn -group. Thompson [27,28] and Flavell [10] proved that G is a solvable group if and only if all the subgroups of G generated by two elements are solvable groups. ...
For a finite group G, we define the subgroup S(G) to be the intersection of the normalizers of the nilpotent residuals of all subgroups of G. Set and define for . The terminal term of this upper series is denoted by . This upper series implies a lot of information on the structure of G. In this paper, we solve several basic problems on S(G). If G=S(G), we call the finite group G an S-group. The new class of S-groups are investigated and some open problems on S-groups are posed. Furthermore, we develop the research on by a new idea and unify some known results.
... Obviously, for the special case A = B = G, the following well-known result of J. Thompson is recovered: (Thompson, [2,3]). A finite group G is soluble if and only if every two-generated subgroup of G is soluble. ...
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.
... The original proofs of all these results depend on part of the classification of finite simple groups. A direct elementary but intricate proof of the existence of 2-variable bases has also been obtained [16]. Theorem 3.2 is a much more general result with a rather straightforward proof but which is again highly dependent on the classification of finite simple groups. ...
... If a group has 2-generator subgroups in a class X which is subgroup-closed and minimal non-X -groups are 2-generator then the group is in X . This is true for the classes of soluble [8,14], supersoluble [6] and nilpotent [9,Satz III.5.2] groups. Minimal non-metacyclic p-groups ( p a prime) have been classified by Blackburn [2,Theorem 3.2]. ...
The aim of this paper is to characterise the finite non-nilpotent groups in which every 2-generator subgroup is metacyclic.
... For example, it is well-known that G is nilpotent if and only if every 2-generator subgroup of G is nilpotent. A deep theorem of Thompson says that G is soluble if and only if every 2-generator subgroup of G is soluble [11] (see also Flavell [4]). A number of recent results reflecting the phenomenon that properties of a finite group are determined by its boundedly generated subgroups can be found in [10,9,2]. ...
The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series of G each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group G is the least number h = h*(G) such that F*h(G) = G, where F*1(G) = F*(G) is the generalized Fitting subgroup, and F*i+1(G) is the inverse image of F* (G/F*i
(G)). In the present paper we prove that if λ(J) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (〈x, x
g
〉) ≤ k for all x, g ∈ G such that 〈x, x
g
〉 is soluble, then h* (G) is k-bounded.
... In [1] ...
... The original proofs of all these results depend on part of the classification of finite simple groups. A direct elementary but intricate proof of the existence of 2-variable bases has also been obtained [16]. Theorem 3.2 is a much more general result with a rather straightforward proof but which is again highly dependent on the classification of finite simple groups. ...
Radicals for Fitting pseudovarieties of groups are investigated from a profinite viewpoint in order to describe Mal’cev products on the left by the corresponding local pseudovariety of semigroups.
... It follows from a theorem of Thompson [19] that the class of all finite soluble groups is 2-recognizable. (See [12] for an elementary proof of this statement.) Hence G is soluble. ...
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.
... Another perspective from which one can look at these results is, loosely speaking, the complexity of their proofs. Thompson proved his solvability criterion (a1) as a corollary to his classification of the minimal simple groups, while Flavell showed in [9] (1995) that it can be proved using elementary methods. On the other hand, the proofs of (a5) and the R(G) membership criteria in (b) rely on the full classification of finite simple groups. ...
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.
... This criterion can itself be viewed as a stronger version of another solvability criterion by Thompson [14,Corollary 2]: a finite group G is solvable if and only if every pair of elements generates a solvable subgroup. We note that Flavell [5] proved the last criterion without using the classification of the minimal simple groups. ...
We observe that a solvability criterion for finite groups, conjectured by Miller [The product of two or more groups, Trans. Amer. Math. Soc. 12 (1911)] and Hall [A characteristic property of soluble groups, J. London Math. Soc. 12 (1937)] and proved by Thompson [Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74(3) (1968)], can be sharpened as follows: a finite group is nonsolvable if and only if it has a nontrivial 2-element and an odd p-element, such that the order of their product is not divisible by either 2 or p. We also prove a solvability criterion involving conjugates of odd p-elements. Finally, we define, via a condition on products of p-elements with p′-elements, a formation Pp,p′, for each prime p. We show that P2,2′ (which contains the odd-order groups) is properly contained in the solvable formation.
... The first motivating result for this paper is the following famous theorem of J. Thompson [22] (see also [7]): a finite group G is solvable if and only if every 2-generated subgroup of G is solvable. ...
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∈G the subgroup of G generated by x and y is solvable. This confirms a conjecture of Flavell. 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.
... The result for two-generated subgroups with respect to the solvability property is provided by a remarkable theorem of Thompson [11,43]: ...
In the paper we pose and discuss new Burnside-type problems, where the role of nilpotency is replaced by that of solvability.
... solvable), so that the pseudovarieties G nil and G sol are both defined by a set of 2-variable pseudo-identities. In the nilpotent case, this is a result of Neumann and Taylor [8] and in the solvable case, it was proved by Thompson [11], see also Flavell [4]. In fact, it is known that there exists an element u nil (a, b) (resp. ...
Let be a pseudovariety of finite groups such that free groups are residually , and let be an injective morphism between finitely generated free groups. We characterize the situations where the continuous extension of between the pro- completions of F(A) and F(B) is also injective. In particular, if is extension-closed, this is the case if and only if and its pro- closure in F(B) have the same rank. We examine a number of situations where the injectivity of can be asserted, or at least decided, and we draw a few corollaries.
... This property was first established by Thompson [32] as a consequence of his monumental classification of finite simple groups whose proper subgroups are solvable, whose proof extends over 400 printed pages and which earned J. G. Thompson the Fields Medal in 1970. A much shorter yet rather involved proof of the 2generator characterization of finite solvable groups has been given by Flavell [16]. ...
this paper with a few minor adjustments. For substructures we take subsets such that whenever an operation is defined on elements of the subset then the resulting value is also in the subset. For a homomorphism, whenever an operation is defined on elements of the domain, the corresponding operation should also be defined on their images and the usual relation (1) should hold. We assume further that there are unary relations in the language which are interpreted in structures so as to form partitions of their universes (into sorts in the language of computer science) and so that all operations take their arguments in one sort and all their values are also of a single sort. Note that this is a nontrivial restriction. It allows us to define products of structures as subsets of the Cartesian product consisting of elements in which all components have the same sort, and then define operations and relations componentwise. Profinite structures are defined as in the case of fully-defined operations and free profinite structures may be constructed by taking projective limits, which in turn are realized as appropriate substructures of products of finite structures
... Since the pseudoidentity u ω = 1 only involves two variables, the validity of this conjecture would entail a result due to Thompson [38] stating that a finite group is solvable if and only if all its 2-generated subgroups are solvable. While Thompson derived this result as a corollary of his complete classification of simple groups whose proper subgroups are solvable, a proof of which extends over 410 published pages, Flavell [18] obtained a direct short and elementary proof of the same corollary. ...
This work gives a new approach to the construction of implicit operations. By considering "higher-dimensional" spaces of implicit operations and implicit operators between them, the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, it is shown how they can be used to deduce from results of Ribes and Zalesski , Margolis, Sapir and Weil, and Steinberg that the pseudovariety of p-groups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups V, if it can be decided whether a finitely generated subgroup of the free group with the pro-V topology is dense, then V is tame.
... The first motivating result for this paper is the following famous theorem of J. Thompson [22] (see also [7]): a finite group G is solvable if and only if every 2-generated subgroup of G is solvable. ...
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.
Every four years leading researchers gather to survey the latest developments in all aspects of group theory. Since 1981, the proceedings of these meetings have provided a regular snapshot of the state of the art in group theory and helped to shape the direction of research in the field. This volume contains selected papers from the 2022 meeting held in Newcastle. It includes substantial survey articles from the invited speakers, namely the mini course presenters Michel Brion, Fanny Kassel and Pham Huu Tiep; and the invited one-hour speakers Bettina Eick, Scott Harper and Simon Smith. It features these alongside contributed survey articles, including some new results, to provide an outstanding resource for graduate students and researchers.
Let G be a finite insoluble group with soluble radical R(G). In this paper we investigate the soluble graph of G, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in G∖R(G), with x adjacent to y if they generate a soluble subgroup of G. Our main result states that this graph is always connected and its diameter, denoted δS(G), is at most 5. More precisely, we show that δS(G)⩽3 if G is not almost simple and we obtain stronger bounds for various families of almost simple groups. For example, we will show that δS(Sn)=3 for all n⩾6. We also establish the existence of simple groups with δS(G)⩾4. For instance, we prove that δS(A2p+1)⩾4 for every Sophie Germain prime p⩾5, which demonstrates that our general upper bound of 5 is close to best possible. We conclude by briefly discussing some variations of the soluble graph construction and we present several open problems.
The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series of G each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group G is the least number h = h* (G) such that F* h (G) = G, where F* 1 (G) = F* (G) is the generalized Fitting subgroup, and F* i+1(G) is the inverse image of F* (G/F*i (G)). In the present paper we prove that if λ(J) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (〈x, xg 〉) ≤ k for allx, g ∈ G such that 〈x, xg 〉 is soluble, then h* (G) is k-bounded.
A comprehensive study of the main research done in polynomial identities over the last 25 years, including Kemer's solution to the Specht problem in characteristic O and examples in the characteristic p situation. The authors also cover codimension theory, starting with Regev's theorem and continuing through the Giambruno-Zaicev exponential rank. The "best" proofs of classical results, such as the existence of central polynomials, the tensor product theorem, the nilpotence of the radical of an affine PI-algebra, Shirshov's theorem, and characterization of group algebras with PI, are presented.
The starting point for this work was the question whether every finite group G contains a two-generated subgroup H such that (H) D (G), where (G) denotes the set of primes dividing the order of G. We answer the question in the affirmative and address the following more general problem. Let G be a finite group and let i.G/ be a property of G. What is the minimum number t such that G contains a t-generated subgroup H satisfying the condition that i(H) D i(G) In particular, we consider the situation where i(G) is the set of composition factors (up to isomorphism), the exponent, the prime graph, or the spectrum of the group G. We give a complete answer in the cases where i(G) is the prime graph or the spectrum (obtaining that t D 3 in the former case and t can be arbitrarily large in the latter case). We also prove that if i(G) is the exponent of G, then t is at most four.
For a commutative ring with identity, say A, its collection of minimal prime ideals is denoted by Min(A). The hull-kernel topology on Min(A) is a well-studied structure. For example, it is known that the hull-kernel topology on Min(A) has a base of clopen subsets, and classifications of when Min(A) is compact abound. Recently, a program of studying the inverse topology on Min(A) has begun. This article adds to the growing literature. In particular, we characterize when Min(A)−1 is Hausdorff. In the final section, we consider rings of continuous functions and supply examples.
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.
Let G be a finite group. It is shown that recent generalizations of two of Thompson's solvability criteria for G can still be improved, and that in a certain sense, these improvements are the best possible.
We characterise the class of finite solvable groups by two-variable identities in a way similar to the characterisation of finite nilpotent groups by Engel identities. Let , and . The main result states that a finite group G is solvable if and only if for some n the identity holds in G. We also develop a new method to study equations in the Suzuki groups. We believe that, in addition to the main result, the method of proof is of independent interest: it involves surprisingly diverse and deep methods from algebraic and arithmetic geometry, topology, group theory, and computer algebra (SINGULAR and MAGMA).
We introduce and study the minimal inner-Σ-Ω-groups and the minimal outer-Σ-℧-groups. Then we give some applications and obtain
some interesting results, including characterizations of nilpotent, supersolvable, solvable, and p-closed groups in terms of the join of two conjugate cyclic subgroups having the same property.
Keywordsnilpotent subgroup–supersolvable subgroup–a pair of conjugate subgroups–minimal inner-Σ-Ω-group
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 a lifting theorem for odd Frattini covers of finite groups. Using
this, we characterize solvable groups and more generally p-solvable groups in
terms of containing a triple of elements of distinct prime power orders with
product 1. This is also related to various questions about Fried's modular
tower program and properties of Hurwitz spaces of covers of curves.
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 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.
Dissertação de Mestrado em Matemática - Fundamentos e Aplicações apresentada à Faculdade de Ciências da Universidade do Porto Um operador implícito n-ário sobre uma pseudovariedade V é um n-uplo de operações implícitas n-árias sobre V. A interpretação de um operador implícito numa álgebra pró-V é a transformação n-ária cujas componentes são as interpretações das operações implícitas que definem esse operador. Abordamos dois temas envolvendo a relação entre a dinâmica de operadores implícitos e a estrutura de grupos finitos. O primeiro desses temas debruça-se sobre operadores implícitos invertíveis; neste caso somos levados a um estudo prévio de algumas propriedades aritméticas do limite projectivo dos anéis dos restos módulo um número natural, o qual é um anel onde o anel dos inteiros está mergulhado. Nomeadamente, mostra mos que um elemento desse limite projectivo é invertível se e só se não for divisível por nenhum primo inteiro. O segundo tema incide sobre os comutadores de Engel. A primeira componente da n-ésima iteração do operador ([x,y],y) é um comutador de Engel. Uma questão que acaba por revelar-se importante é a da escolha de uma definição para o comutador [x,y] entre as opções xyx^(-1)y^(-1) e x^(-1)y^(-1)xy. Se adoptarmos a primeira opção então os grupos finitos onde o operador ([x,y],y) é aperiódico são precisamente os grupos nilpotentes finitos. Se adoptarmos a segunda definição então encontramos exemplos de grupos finitos não nilpotentes onde ([x,y],y) é aperiódico (por exemplo, o grupo simétrico em 3 letras); mostramos que esses grupos são divisíveis pelo grupo simétrico em 3 letras. Entre outras questões relacionadas com o comportamento dinâmico do operador ([x,y],y), destacamos o estudo que fizemos dos grupos diedrais. Referências básicas:J. Almeida, Dynamics of finite semigroups, in Semigroups, Algorithms, Automata and Languages, G. M. S. Gomes, J.-E. Pin, and P. V. Silva, eds., Singapore, 2002, World Scientific, 269-292.F. Grunewald, B. Kuniavskii, D. Nikolova, and E. Plotkin, Two-variable identities in groups and Lie algebras, ...
Let G be a linear group of degree n over a field and let S be any generating set. It is proved that (a) if every pair of products of at most elements of S ∪ S − 1 generates a soluble group then G is soluble and (b) if G is soluble and every pair of products of at most elements of S ∪ S −1 generates a nilpotent group then G is locally nilpotent.
This paper reviews this method of producing implicit operations and explores its usage to obtain pseudoidentity characterizations of various pseudovarieties of groups and semigroups. In particular, the Plotkin conjecture for solvable groups and characterizations of finite nilpotent groups are discussed.
We show that for any finite group G and for any d there exists a word w ∈ Fd such that a d-tuple in G satisfies w if and only if it generates a solvable subgroup. As a corollary, the probability that a word is satisfied in a fixed non-solvable group can be made arbitrarily small; this answers a question of Alon Amit. It also follows that there is no absolute bound in the Baumslag-Pride theorem for the minimal index in a group with at least two more generators than relators of a subgroup that can be mapped homomorphically onto a non-abelian free group.
In the paper we characterize the class of finite solvable groups by two-variable identities in a way similar to the characterization of finite nilpotent groups by Engel identities. More precisely, a sequence of words is called correct if in a group G implies in a group G for all . We are looking for an explicit correct sequence of words such that a group G is solvable if and only if for some n the word is an identity in G. Let , and . The main result states that a finite group G is solvable if and only if for some n the identity holds in G. In the language of profinite groups this result implies that the provariety of prosolvable groups is determined by a single explicit proidentity in two variables. The proof of the main theorem relies on reduction to J.Thompson's list of minimal non-solvable simple groups, on extensive use of arithmetic geometry (Lang - Weil bounds, Deligne's machinery, estimates of Betti numbers, etc.) and on computer algebra and geometry (SINGULAR, MAGMA) .
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.
Let P be a finite p-group, p an odd prime. Using certain versions of p-stability it is shown that there exists a nontrivial characteristic subgroup W in P that is normal in every finite p-stable group G satisfying CG(Op(G)) ≤ Op(G) and . Moreover, W contains every abelian subgroup of P normalized by W.
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.
Let P be a finite p-group, p an odd prime. Using certain versions of p-stability it is shown that there exists a nontrivial characteristic subgroup W in P that is normal in every finite p-stable group G satisfying CG(0p(G)) ≤ Op(G) and P ∈ Sylp(G). Moreover, W contains every abelian subgroup of P normalized by W.