# Characteristically simple and normal fully normalized implies minimal normal

## Statement

If is a Normal fully normalized subgroup (?) of and is characteristically simple as a group, then is a Minimal normal subgroup (?) of .

## Facts used

- Normal upper-hook fully normalized implies characteristic: If are such that is normal in and is fully normalized in , then is characteristic in .

## Proof

**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.