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Fitting classes and lattice formations I
Abstract
A lattice formation is a class of groups whose elements are the direct product of Hall subgroups corresponding to pairwise disjoint sets of primes. In this paper Fitting classes with stronger closure properties involving F-subnormal subgroups, for a lattice formation F of full characteristic, are studied. For a subgroup-closed saturated formation G, a characterisation of the G-projectors of finite soluble groups is also obtained. It is inspired by the characterisation of the Carter subgroups as the N-projectors, N being the class of nilpotent groups.
... It is not true in general that N π -projectors of π -soluble groups are exactly self-N π -Dnormalizing subgroups in N π . Otherwise, N π ∩ S would be either N or S, the class of all soluble groups, by [3,Proposition 4.1]. But we see next that a corresponding result to [3,Theorem 4.2] is still possible. ...
... Otherwise, N π ∩ S would be either N or S, the class of all soluble groups, by [3,Proposition 4.1]. But we see next that a corresponding result to [3,Theorem 4.2] is still possible. That reference provides a corresponding result to our next Theorem 4.14, for finite soluble groups, subgroup-closed saturated formations and associated projectors. ...
Classical results from the theory of finite soluble groups state that Carter subgroups, i.e. self-normalizing nilpotent subgroups, coincide with nilpotent projectors and with nilpotent covering subgroups, and they form a non-empty conjugacy class of subgroups, in soluble groups. This paper presents an extension of these facts to π -separable groups, for sets of primes π , by proving the existence of a conjugacy class of subgroups in π -separable groups, which specialize to Carter subgroups within the universe of soluble groups. The approach runs parallel to the extension of Hall theory from soluble to π -separable groups by Čunihin, regarding existence and properties of Hall subgroups.
... The class N π is a particular case of the so-called lattice formations, which are classes of groups whose elements are direct product of Hall subgroups corresponding to pairwise disjoint sets of primes. With the same flavour as N π -Fitting classes, though within the universe of finite soluble groups, L-Fitting classes, for general lattice formations L of soluble groups, were already defined in [3]. ...
Let π be a set of primes. We show that π-separable groups have a conjugacy class of F-injectors for suitable Fitting classes F, which coincide with the usual ones when specializing to soluble groups.
... 3, Proposition 4.1]. But we see next that a corresponding result to[3, Theorem 4.2] is still possible. That reference provides a corresponding result to our next Theorem 3.13, for finite soluble groups, subgroup-closed saturated formations and associated projectors.As a consequence of Lemma 3.2 we can state the following.Lemma 3.12. ...
Let be a set of primes. We show that -separable groups have a conjugacy class of subgroups which specialize to Carter subgroups, i.e. self-normalizing nilpotent subgroups, or equivalently, nilpotent projectors, when specializing to soluble groups.
Let be a set of primes. We show that -separable groups have a conjugacy class of -injectors for suitable Fitting classes , which coincide with the usual ones when specializing to soluble groups.
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.
Given a lattice formation F of full characteristic, an F - Fitting class is a Fitting class with stronger closure properties involving F -subnormal subgroups. The main aim of this paper is to prove that the associated injectors possess a good behaviour with respect to F -subnormal subgroups.
Dedicated to Professor K. Doerk on his 60th Birthday.
In this paper the subnormal subgroup closed saturated formations of finite soluble groups containing nilpotent groups are fully characterised by means of extensions of well-known properties enjoyed by the formation of all nilpotent groups.
Given a lattice formation F of full characteristic, an F - Fitting class is a Fitting class with stronger closure properties involving F -subnormal subgroups. The main aim of this paper is to prove that the associated injectors possess a good behaviour with respect to F -subnormal subgroups.
In this note we introduce a Fitting class ~=, defined for any Fitting class ~ and set of primes ~, and show how this concept may be used to describe the 3E~J-injectors of a soluble group G, where ~gO denotes the product of the Fitting classes X and ~. Next we investigate the nature of the ~=-injectors of G and show that the natural guess (a product of an ~-injector and a Hall rc'-subgroup), holds good if ~ is a Fischer class (in particular a subgroup closed Fitting class), but more precisely if and only if {~ satisfies a certain condition involving system normalizers. Slight modification of an example in unpublished work of Dark yields a class which fails to satisfy this condition. 2. Preliminaries All groups considered here will belong to the class ~ of finite soluble groups. For each set of primes ~z, ~ will denote the class of finite soluble 7c-groups, and 9l r will denote the class of finite groups of nilpotent length at most r. Our other notation and terminology is standard, terms peculiar to the theory of Fitting classes will be defined as they are introduced. A class of groups ~ is called a Fitting class if
Assume that K = Core G (H) ^ 1. We have that H/K is a self-#-normalizing ^-subgroup of
- G K Moreover
Step 2. We may suppose that Core G (//) = 1.
Assume that K = Core G (H) ^ 1. We have that H/K is a self-#-normalizing
^-subgroup of G/K. Moreover, if H/K < T/K < G/K, then
(H/K) n (T/Kf = (HD T*K)/K = (H n T*)K/K
< (T^YK/K = ((T/K)*)'.
then H is a ^-projector of L and the result is clear by Lemma 2.12
- L If
If L < G, then H is a ^-projector of L and the result is clear by Lemma 2.12.
If L = G, then (HN/N) n (G/AT)* = (HN/N) n (G*/A0 = (// n G*)N/N <
(G«)'N/N = ((G/N)*)'.
Thus H is a ^-projector of G. Then we may suppose that Core
- H By Inductive Hypothesis
By inductive hypothesis, H/K is a ^-projector of G/K. Thus H is a ^-projector
of G. Then we may suppose that Core G (//) = 1.
Fitting classes and lattice formations II', 7. Aust. Math. Soc, to appear. [3] , 'On the lattice of «^-Dnormal subgroups in finite soluble groups
- M Arroyo-Jords
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Soc, to appear.
[3]
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