Finite groups with normally embedded subgroups.

A subgroup H of the finite group G is said to be quasinormally (resp. S-quasinormally) embedded in G if for every Sylow subgroup P of H, there is a quasinormal (resp. S-quasinormal) subgroup K in G such that P is also a Sylow subgroup of K. Groups with certain quasinormally (resp. S-quasinormally) embedded subgroups of prime-power order are studied. For example, if a group G has a normal subgroup H such that G/ H ∈ ℱ and such that for each Sylow subgroup P of H, every member in some ℳ _d( P) is quasinormally embedded in G, then G ∈ ℱ: here ℳ _d( P) is a set of maximal subgroups of P with intersection the Frattini subgroup. [ABSTRACT FROM AUTHOR]