On mp SS-quasinormal subgroups of finite groups

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is said to be an Supplement-Sylow-quasinormal ([Formula: see text]-quasinormal) subgroup of [Formula: see text] if there is a supplement [Formula: see text] of [Formula: see text] to [Formula: see text] such that [Formula: see text] permutes with every Sylow subgroup of [Formula: see text]. In this paper, we present a new criterion for [Formula: see text]-supersolvability of finite groups with [Formula: see text] [Formula: see text]-quasinormal maximal subgroups of Sylow [Formula: see text]-subgroups. As applications, we give some sufficient conditions for a finite group to be [Formula: see text]-nilpotent and supersolvable. Some known results are extended.