Finite groups with sσ-quasinormal subgroups

Let [Formula: see text] be a finite group and [Formula: see text] be some partition of the set [Formula: see text] of all primes. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall [Formula: see text]-set of [Formula: see text] if every member [Formula: see text] of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] for some [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text]. A group [Formula: see text] is said to be a [Formula: see text]-full group if [Formula: see text] possesses a complete Hall [Formula: see text]-set. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-quasinormal in [Formula: see text] if there exists a [Formula: see text]-full subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and for all [Formula: see text], [Formula: see text] permutes with every Hall [Formula: see text]-subgroup of [Formula: see text]. In this paper, we main investigate finite groups all of whose cyclic subgroups of order [Formula: see text] or [Formula: see text] are [Formula: see text]-quasinormal subgroups. Some new criteria for [Formula: see text]-nilpotency and supersolubilities are established.