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Extreme classes of finite soluble groups
... The nilpotent length of G is the smallest integer n such that L,(G) = 1 (or, equivalently, F n (G) = G) and will be denoted by /(G). As in [1] the series {<&{G)} is denned by O.CG^F.^CG) = OCC/F^CC)), i = 1 , 2 , -. ...
... Following [1], we say a group G is extreme if F f (G)/<J>,(G) is a chief factor of G for each i = 1,2, •• ,/(G). From [ l j Theorem 2.8 we have LEMMA 2.1. ...
... The number of complemented chief factors in a chief series of a group G will be denoted by c(G). (Lemma 2.6 of [1] shows that this number is independent of the particular chief series.) The following characterization of extreme groups is given in [1] Theorem 2.9 (hi). ...
Throughout this paper a “group” will mean a “finite soluble group”.
... The classes 2t, 9t, of finite Abelian and finite nilpotent groups respectively, are star classes, being formations satisfying the extra properties. U, the class of finite supersoluble groups, is a star class by a result of Carter, Fischer and Hawkes [2]. (In [2] groups are soluble; however the proof of Corollary 2 below shows that a finite group is necessarily soluble if every pair of its elements generates a supersoluble subgroup.) ...
... U, the class of finite supersoluble groups, is a star class by a result of Carter, Fischer and Hawkes [2]. (In [2] groups are soluble; however the proof of Corollary 2 below shows that a finite group is necessarily soluble if every pair of its elements generates a supersoluble subgroup.) A proof by induction using another result from [2] -see [3, 6.15 on p. 523] -shows that product classes S P l S P 2 . . . ...
... Let ^3* be the subclass of ^J n of non-Frobenius groups with minimally non-Abelian stabilisers. Since a minimally non-Abelian group is 2-generator the upshot of (1), this paragraph and the last is that (2) there are groups in ^3* with arbitrarily large kernels. ...
A special case of the main result is the following. Let G be a finite, non-supersoluble group in which from arbitrary subsets X, Y of cardinality n we can always find x ∈ X and y ∈ Y generating a supersoluble subgroup. Then the order of G is bounded by a function of n. This result is a finite version of one line of development of B.H. Neumann's well-known and much generalised result of 1976 on infinite groups.
... Let $ be a class of finite soluble groups with the properties: (1) ^ is a Fitting class (i.e. normal subgroup closed and normal product closed) and (2) if N ^ H Ĝ e %, N <i G and HjN is a /?-group for some prime p, then He^. Then % is called a Fischer class. ...
... In this section we will give simple characterizations of partially ^-complemented chief factors of a group G, which we shall need in the next section. We begin 395 A. Makan [2] with two examples. That a chief factor of G may be partially ^-complemented without being ^-complemented is shown by the following example. ...
... Consider any two chief series of G/P G . By Lemma 2.6 [2] there is a 1-1 correspondence between complemented chief factors in the above series, corresponding chief factors being G-isomorphic. In particular there is such correspondence between complemented /?-chief factors of G/P G . ...
Let name be a class of finite soluble groups with the properties: (1) is a Fitting class (i.e. normal subgroup closed and normal product closed) and (2) if N ≦ H ≦ G ∈ , N ⊲ G and H/N is a p -group for some prime p , then H ∈ . Then is called a Fischer class. In any finite soluble group G , there exists a unique conjugacy class of maximal -subgroups V called the -injectors which have the property that for every N◃◃G, N ∩ V is a maximal -subgroup of N [3]. 3. By Lemma 1 (4) [7] an -injector V of G covers or avoids a chief factor of G. As in [7] we will call a chief factor -covered or -avoided according as V covers or avoids it and -complemented if it is complemented and each of its complements contains some -injector. Furthermore we will call a chief factor partially -complemented if it is complemented and at least one of its complements contains some -injector of G .
... It seems that Gaschütz's work was the driving force for the further study of finite soluble groups at that time. Roger went on to collaborate on one further article [31] on soluble groups. Trevor Hawkes relates: "In 1966 I got an invitation from Reinhold Baer to spend a couple of weeks in Frankfurt and met Bernd Fischer there. ...
... Roger Carter and Sandy Green were already there, and Brian Hartley subsequently joined us, initially commuting from Cambridge on a motorbike. While Fischer was on an extended visit to Warwick, he, Roger and I wrote a joint paper, called "Extreme Classes of Finite Soluble Groups" [31]. Those were exciting times when the Mathematics Institute was in an isolated house on Gibbet Hill, more than a mile from the central Warwick Campus, and my office was a former bathroom. ...
Roger Carter (1934--2022) was a very well known mathematician working in algebra, representation theory and Lie theory. He spent most of his mathematical career in Warwick. Roger was a great communicator of mathematics: the clarity, precision and enthusiasm of his lectures delivered in his beautiful handwriting were hallmark features recalled by numerous students and colleagues. His books have been described as marvelous pieces of scholarship and service to the general mathematical community. We both met Roger early in our careers, and were encouraged and influenced by him~ -- ~and his lovely sense of humour. This text is our tribute, both to his mathematical achievements, and to his kindness and generosity towards his students, his colleagues, his collaborators, and his family.
... Recall [3] that a class of groups H is called extreme if it is a saturated homomorph such that from G/K ∈ H and K is a unique minimal normal subgroup of G it follows that G ∈ H. Extreme classes of groups play an important role in the structural study of local formations (see [3,20]). ...
... Recall [3] that a class of groups H is called extreme if it is a saturated homomorph such that from G/K ∈ H and K is a unique minimal normal subgroup of G it follows that G ∈ H. Extreme classes of groups play an important role in the structural study of local formations (see [3,20]). ...
In this paper, the classes of groups with given systems of -subnormal subgroups are studied. In particular, it is showed that if and are a saturated homomorph and a hereditary saturated formation, respectively, then the class of groups whose -subgroups are all -subnormal is a hereditary saturated formation. As corollaries, some known results about supersoluble groups, classes of groups with -subnormal cyclic primary and Sylow subgroups are obtained. Also the new characterization of the class of groups whose extreme subgroups all belong , where is a hereditary saturated formation, is obtained.
... Next we give the details of the twisted wreath product construction and its associated notation (cf. [6,1,15.10]). Let X and G be groups. ...
... Each of the following is necessary and sufficient for a nontrivial group G to belong to (a) G has a unique chief series and all its chief factors are complemented; (b) The lower nilpotent series of G is a chief series; (c) The upper nilpotent series of G is a chief series; (d) G is extreme (cf. Definition 3 of [1]) and G has prefrattini subgroups of order 1. ...
The group construction sometimes known as the twisted wreath product is used here to answer two questions in the theory of finite, soluble groups: first to show that an arbitrary finite, soluble group may be embedded as a sub group of а group whose upper nilpotent series is a chief series; second to construct an А-group whose Carter subgroup is “small” relative to its nilpotent length.
... Much of this is inspired by corresponding work in group theory in [5] and [4], but there are significant differences encountered in the Lie case. ...
The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-}. To facilitate this we investigate solvable Lie algebras of nilpotent length k, and of nilpotent length , and {\em extreme} Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non- Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index .
... Embed A in V = Q 0 ~ A by identifying A and 1 @ A. V is a QG-module containing an ascending series of QG-submodules whose factors have Q-dimension at most 1. Also, V is completely reducible as QG-module (Extension of Maschke's Theorem, Ref. [7], IV.8.j) and thus V is a direct sum of irreducible QG-modules of Q-dimension 1. Let Vi be the homogeneous components of V as QG-module and put Ai = A n Vi. ...
If G is a subgroup of GL (n, F) G has paraheight at most w + [log, n!]. If G is a subgroup of GL (n, R) where R is a finitely generated integral domain then G has finite Paraheight.
... Much of this is inspired by corresponding work in group theory in [5] and [4], but there are significant differences encountered in the Lie case. ...
The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-}. To facilitate this we investigate solvable Lie algebras of nilpotent length k, and of nilpotent length , and {\em extreme} Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non- Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index .
... Much of this is inspired by corresponding work in group theory in [5] and [4], but there are significant differences encountered in the Lie case. ...
... The fact that supersolubility is bigenetic in finite groups is a theorem of Carter, Fischer and Hawkes [3,Theorem 4.8] (for another proof see [18,Lemma 3.5]). ...
Let be a class and p a property of groups. We say that p is a bigenetic property of p-groups (or more simply, p is bigenetic in p-groups) if an p-group G has the property p whenever all two-generator subgroups of G have p.
... From Corollary 1 with A = B = G we deduce that for each prime p dividing the order of G, G f (p) centralizes any p-chief factor of G, which implies G ∈ F . 2 Remark. We point out that Corollary 2 follows also easily from a result of Carter, Fischer and Hawkes [8,Theorem 4.2]; see also [11,VII,6.15]: This result states in particular that in the universe of all finite soluble groups a saturated formation is 2-recognizable if it has a local definition of 2-recognizable 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.
... Clearly, Z -j t K X Z/Y is minimal normal in X. If Z is complemented in X, then * G E c % C g % by (2), while X G 2 % by (3) in case that Z < <&(X). Therefore, in any case we have G = X/Y E Q{X) C Q% = X, the desired contradiction. ...
We define and investigate H-prefrattini subgroups for Schunck classes H of finite soluble groups, and solve a problem of Gaschütz concerning the structure of H-prefrattini groups for H = {1}.
... Let HjK be a complemented chief factor of a group G and M one of the maximal subgroups of G complementing it. Writing C = C G (H/K) it is well known that Core(M) = M O C and that CjCC\M is a chief factor * The closure operation of taking finite direct products is denoted by Da; the other closure operations mentioned are defined in [1]. A more detailed analysis of their properties appears in [4]. ...
In his Canberra lectures on finite soluble groups, [3], Gaschütz observed that a Schunck class (sometimes called a saturated homomorph) is { Q , E φ , D 0 }-closed but not necessarily R 0 closed(*). In Problem 7·8 of the notes he then asks whether every { Q , E φ , D 0 }-closed class is a Schunck class. We show below with an example † that this is not the case, and then we construct a closure operation R 0 satisfying D o < r o < R o such that is a Schunck class if and only if = { Q E φ, R o }. In what follows the class of finite soluble groups is universal. Let B denote the class of primitive groups. We recall that a Schunck class is one which satisfies: (a) = Q , and (b) contains all groups G such that Q ( G ) ∩ B ⊆.
... At the end of our considerations we will prove a general result in this direction. For it the following definition of [2] seems to be helpful: DEFINITION. Let H and ~-be classes of groups, o~-is called H-complete if the following is true: If all H-subgroups of a group G belong to ~-, then G itself belongs to o~. ...
Given two classes and of finite groups, is said to have property for whenever contains every -group G expressible as the product of some subgroups such that the groups lie in for all . This article describes all Z -saturated -closed formations and Fischer formations of solvable groups with property . In particular, the set of all such formations coincides with the set of hereditary Shemetkov formations in the class of all finite solvable groups. We describe the hereditary saturated formations with every saturated subformation having property for .
In 1968 Hawkes proved that for every soluble finite group the intersection of the Fitting subgroup with a -crucial maximal subgroup is its Fitting subgroup where is a skeletal class of soluble groups. He suggested that this result might spring from a more general one. We prove that all hereditary Fitting classes of soluble groups such that for every soluble finite group the intersection of the -radical with an -crucial maximal subgroup is its -radical, where is the least by inclusion skeletal class of soluble groups, are exactly classes of all -nilpotent soluble finite groups.
The article contains results about invariants of solvable groups with given structure of Sylow subgroups and information about the nilpotent π-length of π-solvable groups. Open questions are formulated.
If the quotient group of a group G modulo its hypercenter is a T-group, we call G a -group. We classify all finite groups which are not -groups but all their proper subgroups are, and compare this with the situation for supersolvable groups, T-groups, and PST-groups.
All groups considered are finite. In recent years a number of generalizations of the classic Jordan-Hölder Theorem have been obtained (see [7], Theorem A.9.13): in a finite group G a one-to-one correspondence as in the Jordan-Holder Theorem can be defined preserving not only G -isomorphic chief factors but even their property of being Frattini or non-Frattini chief factors. In [2] and [13] a new direction of generalization is presented: the above correspondence can be defined in such a way that the corresponding non-Frattini chief factors have the same complement (supplement).
Let
be a saturated formation. We describe minimal non-
-, minimal non-
-, and minimal non-metabelian 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.
In 1968 Carter, Fisher, and Hawkes developed the method of extremal classes to study local formations and their critical groups
in the class of finite solvable groups. In the present paper the problem of Shemetkov on the extension of this method to arbitrary
finite groups is solved.
We prove here that the (saturated) formation generated by a finite soluble group has only finitely many (saturated) subformations. This answers a question asked by Professor W. Gaschütz. Some partial results are also given in the case of a formation generated by an arbitrary finite group.
Let G = H0 > H1 > … > Hr = 1 and G = K0 > K1 > … > Kr =1 be two chief series of the finite soluble group G. Suppose Mi complements Hi/Hi+1. Then Mi also complements precisely one factor Kj/Kj+1, of the second series, and this Kj/Kj+1 is G-isomorphic to Hi/Hi+1. It is shown that complements Mi can be chosen for the complemented factors Hi/Hi+1 of the first series in such a way that distinct Mi complement distinct factors of the second series, thus establishing a one-to-one correspondence between the complemented factors of the two series. It is also shown that there is a one-to-one correspondence between the factors of the two series (but not in general constructible in the above manner), such that corresponding factors are G-isomorphic and have the same number of complements.
LetGbe a polycyclic group. We prove that if the nilpotent length of each finite quotient ofGis bounded by a fixed integern, then the nilpotent length ofGis at mostn. The casen=1 is a well-known result of Hirsch. As a consequence, we obtain that if the nilpotent length of each 2-generator subgroup is at mostn, then the nilpotent length ofGis at mostn. A more precise result in the casen=2 permits us to prove that if each 3-generator subgroup is abelian-by-nilpotent, thenGis abelian-by-nilpotent. Furthermore, we show that the nilpotent length ofGequals the nilpotent length of the quotient ofGby its Frattini subgroup.
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.
A finite group is supersoluble if and only if its poset of subgroups satisfies the Jordan–Dedekind chain condition.
An element x of a group G is called Q-central if there exists a central chief factor H/K of G such that x ∈ H\K. It is proved that a finite group G is p-nilpotent if and only if every element in Gp \Φ(Gp) is Q-central. We will adapt Doerk and Hawkes [Doerk, K., Hawkes, T. (1992). Finite Soluble Groups. Berlin–New York: Walter de Gruyter, pp. 892] for notations and basic results.
Ascending series. Let G be a group and λ an ordinal number. An ascending series of G of type λ is a set of subgroups Gα of G, defined for all α ≤ λ, and such that (i) G0 = 1, Gλ = G; (ii)Gα is a proper normal subgroup of Gα+1 for all α < λ (iii) for all limit ordinals μ ≤ λ.(Received November 13 1962)
On the p-length of p-soluble groups and reduction 13. SCHMIDT, 0. J. Uber Gruppen deren siimtliche Teiler spezielle sind
- P Hall
- G And Higman
, HALL, P. AND HIGMAN, G. On the p-length of p-soluble groups and reduction 13. SCHMIDT, 0. J. Uber Gruppen deren siimtliche Teiler spezielle sind. Rec. Moth. Moscow, 31 (1924), 366-372.
= 2n + 1. Critical groups for f&(n) may also be characterized in terms of the auto-morphism groups induced on the Frattini chief factors
- G Hence
- Z Fzn+l
Hence G = Fzn+l and Z(G) = 2n + 1. Critical groups for f&(n) may also be characterized in terms of the auto-morphism groups induced on the Frattini chief factors. THEOREM 5.
*) in this case also, and the proof is complete. Our next result gives additional information about the groups S
- H Thus
Thus H/K satisfies (*) in this case also, and the proof is complete. Our next result gives additional information about the groups S,. AND HAWKES O,(M,) = 1 since this centralizes Fl, and so Z,(M#I,(M,))
Construction of Formations in Finite Soluble Groups
- Lubeseder
Il. LUBESEDER, U. "Construction
of Formations
in Finite Soluble Groups."
Thesis,
University
of Kiel, Germany, 1963.
Uber Gruppen deren siimtliche Teiler spezielle sind
SCHMIDT, 0. J. Uber Gruppen deren siimtliche Teiler spezielle sind. Rec. Moth.
Moscow, 31 (1924), 366-372.
Über Gruppen deren sämtliche Teiler spezielle sind
- Schmidt