# Every group is normal fully normalized in its holomorph

From Groupprops

## Statement

Suppose is a group. Denote by the holomorph of ; in other words, we have:

with the natural action of the automorphism group on . Then, is a normal fully normalized subgroup of :

- is normal in : Every inner automorphism of restricts to an automorphism of .
- is fully normalized in : Every automorphism of extends to an inner automorphism in .