Sylow normalizer implies abnormal

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Sylow normalizer (?)) must also satisfy the second subgroup property (i.e., Abnormal subgroup (?)). In other words, every Sylow normalizer of finite group is a abnormal subgroup of finite group.
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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Sylow subgroup (?)) must also satisfy the second subgroup property (i.e., Subgroup with abnormal normalizer (?)). In other words, every Sylow subgroup of finite group is a subgroup with abnormal normalizer of finite group.
View all subgroup property implications in finite groups | View all subgroup property non-implications in finite groups | View all subgroup property implications | View all subgroup property non-implications

Statement

The normalizer of any Sylow subgroup in a finite group is an abnormal subgroup.

Facts used

  1. Sylow implies pronormal
  2. Normalizer of pronormal implies abnormal

Proof

The proof follows directly by combining facts (1) and (2).