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 in [Formula: see text] if [Formula: see text] for all [Formula: see text]; [Formula: see text] is called weakly [Formula: see text]-subgroup in [Formula: see text] if it has a normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] is an [Formula: see text]-subgroup in [Formula: see text]. In this paper, we present some sufficient conditions for a group to be [Formula: see text]-nilpotent under the assumption that certain subgroups of fixed prime power orders are weakly [Formula: see text]-subgroups in [Formula: see text]. The main results improve and extend new and recent results in the literature.
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