# Isomorph-normal coprime automorphism-invariant of Sylow implies weakly closed

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a group of prime power order. That is, it states that in a Group of prime power order (?), every subgroup satisfying the first subgroup property (i.e., Isomorph-normal coprime automorphism-invariant subgroup (?)) must also satisfy the second subgroup property (i.e., Sylow-relatively weakly closed subgroup (?)). In other words, every isomorph-normal coprime automorphism-invariant subgroup of group of prime power order is a Sylow-relatively weakly closed subgroup of group of prime power order.

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## Contents

## Statement

Suppose is a group of prime power order (where the underlying prime is ) and is an isomorph-normal coprime automorphism-invariant subgroup of . In other words, we have the following:

- is isomorph-normal in : Any subgroup of isomorphic to is normal in .
- is coprime automorphism-invariant in : Any -automorphism of leaves invariant.

Then, for any finite group containing as a -Sylow subgroup, is weakly closed in with respect to .

## Related facts

- Isomorph-normal characteristic of WNSCDIN implies weakly closed
- Isomorph-containing iff weakly closed in any ambient group
- Isomorph-normal coprime automorphism-invariant implies weakly closed for any fusion system

## Facts used

- Coprime automorphism-invariant normal subgroup of Hall subgroup is normalizer-relatively normal
- Isomorph-normal implies conjugation-invariantly relatively normal in any ambient group
- Sylow implies WNSCDIN
- WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

## Proof

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