Abstract
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 M d ( P ) denote a family of maximal subgroups P 1 , … , P d of P satisfying ⋂ i = 1 d ( P i ) = Φ ( P ) , the Frattini subgroup of P. Assume that the group G is p-solvable and every member of some fixed M d ( P ) is SS-quasinormal in G, then G is p-supersolvable.
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