Hall subgroups and \(\pi\)-separable groups
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 \(q\in\pi'\).
scientific article; zbMATH DE number 1088611
scientific article