Normal upper-hook fully normalized implies characteristic
If are groups, with normal in , and fully normalized in , then is characteristic in .
- Characteristic upper-hook AEP implies characteristic
- Fully invariant upper-hook EEP implies fully invariant
- 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
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.