Abstract
Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text] for all [Formula: see text]. We say that [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text] if [Formula: see text] has a normal subgroup [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 Sylow [Formula: see text]-subgroup of [Formula: see text]. We fix a subgroup of [Formula: see text] of order [Formula: see text] with [Formula: see text] and study the structure of [Formula: see text] under the assumption that every subgroup of [Formula: see text] of order [Formula: see text] [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text]. Our results improve and generalize several recent results in the literature.
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