Abstract
Let Open image in new window be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are presented: (1) G ∈ Open image in new window if and only if there is a normal subgroup H such that G/H ∈ Open image in new window and every maximal subgroup of all Sylow subgroups of H is either c-normal or S-quasinormally embedded in G. (2) G ∈ Open image in new window if and only if there is a normal subgroup H such that G/H ∈ Open image in new window and every maximal subgroup of all Sylow subgroups of F*(H), the generalized Fitting subgroup of H, is either c-normal or S-quasinormally embedded in G. (3) G ∈ Open image in new window if and only if there is a normal subgroup H such that G/H ∈ Open image in new window and every cyclic subgroup of F*(H) of prime order or order 4 is either c-normal or S-quasinormally embedded in G.
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