Abstract
Suppose that [Formula: see text] is a finite group and [Formula: see text] is a subgroup of [Formula: see text]. [Formula: see text] is said to be an [Formula: see text]-quasinormal subgroup of [Formula: see text] if there is a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] permutes with every Sylow subgroup of [Formula: see text]. In this note, we fix in every non-cyclic Sylow subgroup [Formula: see text] of [Formula: see text] some subgroup [Formula: see text] satisfying [Formula: see text] and study the [Formula: see text]-nilpotency of [Formula: see text] under the assumption that every subgroup [Formula: see text] of [Formula: see text] with [Formula: see text] is [Formula: see text]-quasinormal in [Formula: see text]. The Frobenius theorem is generalized.
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