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Hall subgroups and \(\pi\)-separable groups. (English) Zbl 0889.20011

Let \(E_\pi\) denote the collection of groups that have Hall \(\pi\)-subgroups for a set \(\pi\) of primes. The author’s abstract: Let \(G\) be a finite group and \(\pi\) a set of primes. In this paper, some new criteria for \(\pi\)-separable groups and \(\pi\)-solvable groups in terms of Hall subgroups are proved.
The main results are the following: Theorem. \(G\) is a \(\pi\)-separable group if and only if (i) \(G\) satisfies \(E_\pi\) and \(E_{\pi'}\); (ii) \(G\) satisfies \(E_{\pi\cup(q)}\) and \(E_{\pi'\cup(p)}\) for all \(p\in\pi\), \(q\in\pi'\). Theorem. \(G\) is a \(\pi\)-separable group if and only if (i) \(G\) satisfies \(E_\pi\) and \(E_{\pi'}\); (ii) \(G\) satisfies \(E_{p,q}\) for all \(p\in\pi\) and \(q\in\pi'\).

MSC:

20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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References:

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