On solubility and supersolubility of some classes of finite groups

Abstract  Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let H _sG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H ⩽ H _sG and HT = C. Our main result is the following Theorem A   A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgroup F*(G) of G, at least one of the following holds:

(1)	Every maximal subgroup of P is S-embedded in G.
(2)	Every cyclic subgroup H of P with prime order or order 4 (if P is a non-abelian 2-group and H ⊈ Z ∞ (G)) is S-embedded in G.

This work was supported by National Natural Science Foundation of China (Grant No. 10771180)