Sylow implies pronormal

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., Pronormal subgroup (?)). In other words, every Sylow subgroup of finite group is a pronormal subgroup of finite group.
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Statement

Any Sylow subgroup of a finite group is pronormal.

Definitions used

Sylow subgroup

Further information: Sylow subgroup

Pronormal subgroup

Further information: Pronormal subgroup

Related facts

Stronger facts

• Sylow of normal implies pronormal

Corollaries

• Sylow normalizer implies abnormal
• Sylow normalizer implies upward-closed self-normalizing
• Sylow implies intermediately subnormal-to-normal

Facts used

1. Sylow implies intermediately isomorph-conjugate
2. Intermediately isomorph-conjugate implies pronormal

Proof

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