# Sylow of normal implies pronormal

From Groupprops

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 of normal subgroup (?)) must also satisfy the second subgroup property (i.e., Pronormal subgroup (?)). In other words, every Sylow subgroup of normal subgroup of finite group is a pronormal subgroup of finite group.

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This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Sylow subgroup (?) and Normal subgroup (?)), to another known subgroup property (i.e., Pronormal subgroup (?))

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## Statement

Any Sylow subgroup of a normal subgroup of a group is pronormal.

## Facts used

- Sylow implies intermediately isomorph-conjugate
- Intermediately isomorph-conjugate of normal implies pronormal

## Proof

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