Characteristically simple and normal fully normalized implies minimal normal
- Normal upper-hook fully normalized implies characteristic: If are such that is normal in and is fully normalized in , then is characteristic in .
Given: A group , a characteristically simple, normal and fully normalized subgroup .
To prove: is a minimal normal subgroup of .
Proof: By assumption is normal, so it suffices to show that any normal subgroup of contained in is either trivial or equal to . Let's do this.
By fact (1), is characteristic in . Since is characteristically simple, we see that must be either equal to , or trivial, completing the proof.