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Sufficient conditions for supersolubility of finite groups. (English) Zbl 0928.20020
A subgroup \(H\) of a finite group \(G\) is \(S\)-quasinormal in \(G\) if it permutes with each Sylow subgroup of \(G\); a subgroup \(K\) of \(G\) is \(S\)-quasinormally embedded in \(G\) if each Sylow subgroup of \(K\) is also a Sylow subgroup of some \(S\)-quasinormal subgroup of \(G\). The authors prove: Theorem 1. If each maximal subgroup of the Sylow subgroups of \(G\) is \(S\)-quasinormally embedded in \(G\), then \(G\) is supersolvable. Theorem 2. If \(G\) is a solvable group with normal subgroup \(H\) such that \(G/H\) is supersolvable and all maximal subgroups of the Sylow subgroups of the Fitting subgroup of \(H\) are \(S\)-quasinormally embedded in \(G\), then \(G\) is supersolvable.

20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20E28 Maximal subgroups
20D25 Special subgroups (Frattini, Fitting, etc.)
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
Full Text: DOI
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