# Normal upper-hook fully normalized implies characteristic

From Groupprops

## Statement

If are groups, with normal in , and fully normalized in , then is characteristic in .

## Related facts

- Characteristic upper-hook AEP implies characteristic
- Fully invariant upper-hook EEP implies fully invariant

### Applications

- Left transiter of normal is characteristic can be derived from the statement of this page, along with the fact that every group is normal fully normalized in its holomorph.
- Characteristically simple and normal fully normalized implies minimal normal

## Proof

**Given**: are groups, with normal in , and fully normalized in .

**To prove**: is characteristic in .

**Proof**: Let be any automorphism of . We need to show that .

Since is fully normalized in , there exists such that equals conjugation by . Since is normal in , conjugation by leaves invariant, so , completing the proof.