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Extension of Carter subgroups in -separable groups
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Abstract
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.
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In this paper, we complete the classification of Hall subgroups in finite almost simple groups. We also prove that a finite group satisfies D π if and only if all its composition factors satisfy D π .
LetGbe a finite group and π a set of primes. In this paper, some new criteria for π-separable groups and π-solvable groups in terms of Hall subgroups are proved. The main results are the following:Theorem.G is aπ-separable group if and only if(1)G satisfies Eπand Eπ′;(2)G satisfies Eπ∪{q}and Eπ′∪{p}for all p∈π,q∈π′.Theorem.G is aπ-separable group if and only if(1)G satisfies Eπand Eπ′;(2)G satisfies Ep,qfor all p∈πand q∈π′.
Summary Dp-property (p=set of primes) in finite groups is not in general inherited by subgroups. In this paper, as evidence in favor of the following conjecture (F. Gross): (o) If a finite group G satisfies Dp then its normal subgroups satisfy Dp-property as well. the Author shows that if the Dp and the Dp-properties (p'=set of the primes not in p) hold together in a finite group G, then both are inherited by the normal subgroups of G. As a corollary, the characterization of the groups satisfying both the properties Dp and Dp' is given in terms of the composition factors.
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.
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