We investigate a finite groupGwith{\mathfrak{F}}-subnormal Sylow subgroups, where{\mathfrak{F}}is a subgroup-closed formation and{\mathfrak{A}_{1}\mathfrak{A}\subseteq\mathfrak{F}\subseteq\mathfrak{N}% \mathcal{A}}. We prove thatGis soluble and the derived subgroup of each metanilpotent subgroup is nilpotent. We also describe the structure of groups in which every Sylow subgroup is{\mathfrak{F}}-subnormal or{\mathfrak{F}}-abnormal.
Let H be a subgroup of a group G. The permutizer P G (H) is the subgroup generated by all cyclic subgroups of G which permute with H. A subgroup H of a group G is strongly permutable in G if P U (H) = U for every subgroup U of G such that H ≤ U ≤ G. We investigate groups with P-subnormal or strongly permutable Sylow and primary cyclic subgroups. In particular, we prove that groups with all strongly permutable primary cyclic subgroups are supersoluble.
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