Characteristically simple and normal fully normalized implies minimal normal

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Statement

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

Facts used

  1. Normal upper-hook fully normalized implies characteristic: If K \le H \le G are such that K is normal in G and H is fully normalized in G, then K is characteristic in H.

Proof

Given: A group G, a characteristically simple, normal and fully normalized subgroup H.

To prove: H is a minimal normal subgroup of G.

Proof: By assumption H is normal, so it suffices to show that any normal subgroup K of G contained in H is either trivial or equal to H. Let's do this.

By fact (1), K is characteristic in H. Since H is characteristically simple, we see that K must be either equal to H, or trivial, completing the proof.