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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.

##### MSC:
 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
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##### References:
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