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Fitting classes and lattice formations II
Abstract
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 manner, notice that an ^"-Fitting class is also a Fitting class, as the lattice formation J*" contains jV. In a forthcoming paper [2], the desired behaviour of the associated injectors, with respect to ^"-subnormal (and ^"-Dnormal) subgroups, is obtained. In fact, this property characterizes ^"-Fitting classes. ...
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.
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advances made, give a new approach to some classic topics, shed light on some fundamental facts that still remain unpublished and present some new subjects of research in the theory of classes of finite, not necessarily solvable, groups.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 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.
then statement (ii) is clear because every G, is &subnormal in G. For the converse, argue as in the proof of
- Proof If V E I N J
PROOF. If V e I n j ^ G ), then statement (ii) is clear because every G, is &subnormal in G.
For the converse, argue as in the proof of [8, VIII, Proposition 2.12] taking Corollary 3.10 into account.
@BULLET
References
S 1 '-Pronormale Untergruppen von endlich auflosbarer Gruppen (Diplomarbeit
- N Miiller
N. Miiller, S 1 '-Pronormale Untergruppen von endlich auflosbarer Gruppen (Diplomarbeit, Johannes Gutenberg-Universitat Mainz, 1985).