The influence of SS-quasinormality of some subgroups on the structure of finite groups

The following concept is introduced: a subgroup H of the group G is said to be SS-quasinormal (Supplement-Sylow-quasinormal) in G if H possesses a supplement B such that H permutes with every Sylow subgroup of B. Groups with certain SS-quasinormal subgroups of prime power order are studied. For example, fix a prime divisor p of |G| and a Sylow p-subgroup P of G, let d be the smallest generator number of P and Md(P) denote a family of maximal subgroups P1,…,Pd of P satisfying , the Frattini subgroup of P. Assume that the group G is p-solvable and every member of some fixed Md(P) is SS-quasinormal in G, then G is p-supersolvable.