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If all subgroups of F ( E ) of prime order or order 4 are c-normal in G, then G is supersoluble.
Then (Gin fancyscript{F}) if and only if every cyclic subgroup of any Sylow subgroup of (F^(E)) of prime order or order 4 is (s -semipermutable in (G).
Then (Gin fancyscript{F}) if and only if every cyclic subgroup of any Sylow subgroup of (F^(H)) of prime order or order 4 is (ss -quasinormal in (G).
Then (Gin fancyscript{F}) if and only if every cyclic subgroup of any Sylow subgroup of (F^(E)) of prime order or order 4 is (s -quasinormal in (G).
Suppose that for every non-cyclic Sylow subgroup P of E, every cyclic subgroup H of P of prime order or order 4 (if P is a non-Abelian 2-group and H ⊈ Z ∞ ( G ) ) is λ-supplemented in G provided that H does not possess any supersoluble supplement in G. Then G is supersoluble.
Let ℱ be a saturated formation containing all supersoluble groups and G be a group with a normal subgroup E such that G / E ∈ F. If every cyclic subgroup of F ∗ ( E ) of prime order or order 4 is S-permutable in G, then G ∈ F. Corollary 4.11 [21].
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com