# Automorph-conjugacy is normalizer-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., automorph-conjugate subgroup) satisfying a subgroup metaproperty (i.e., normalizer-closed subgroup property)

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## Statement

Suppose is a group and is an automorph-conjugate subgroup of , then (the normalizer of in ) is also an automorph-conjugate subgroup of .

## Related facts

### Generalizations

- Automorphism-based relation-implication-expressible implies normalizer-closed: This is a general formalism result stating that any subgroup property that can be expressed in a given format is closed under taking normalizers.

## Proof

### Proof idea

The key idea behind the proof is that *taking the normalizer* commutes with automorphisms. In other words, for any automorphism , we have .

### Proof details

**Given**: A group , an automorph-conjugate subgroup .

**To prove**: is an automorph-conjugate subgroup: given any automorphism of , and are conjugate.

**Proof**: . By assumption, is automorph-conjugate, so there exists a such that . Thus, , and thus, and are conjugate by .