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A note on minimal subgroups of finite groups. (English) Zbl 0856.20015

Let \(G\) be a finite group and \(\psi(G)=\langle x\mid x\in G\), \(|x|\) is a prime number or \(|x|=4\rangle\). A subgroup \(H\) of a group \(G\) is \(\pi\)-quasinormal in \(G\) if \(H\) permutes with every Sylow subgroup of \(G\). An element \(x\in G\) is \(\pi\)-quasinormal in \(G\) if \(\langle x\rangle\) is a \(\pi\)-quasinormal subgroup of \(G\).
Let \(\mathfrak F\) be a saturated formation containing the formation of all supersoluble groups. Let \(G\) be a group with a normal subgroup \(H\) such that \(G/H\in{\mathfrak F}\). It is proved that if every generator of \(\psi(H)\) is \(\pi\)-quasinormal in \(G\) or if \(H\) has a Sylow tower of supersoluble type and for each Sylow subgroup \(P\) of \(H\) every generator of \(\psi(P)\) is \(\pi\)-quasinormal in \(N_G(P)\), then \(G\in{\mathfrak F}\).

MSC:

20D25 Special subgroups (Frattini, Fitting, etc.)
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
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