Finite groups with some subgroups of Sylow subgroups weakly ℋC-embedded

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called weakly [Formula: see text]-embedded in [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text] where [Formula: see text] is the normal closure of [Formula: see text] in [Formula: see text] For each prime [Formula: see text] dividing the order of [Formula: see text] let [Formula: see text] be a non-cyclic Sylow [Formula: see text]-subgroup of [Formula: see text] We fix a [Formula: see text]-power integer [Formula: see text] with [Formula: see text], and study the structure of [Formula: see text] under the assumption that each subgroup of order [Formula: see text] [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text] Some new results about the [Formula: see text]-nilpotency and supersolvability of [Formula: see text] are obtained.