Characteristically simple and non-abelian implies automorphism group is complete

Statement

If $G$ is a Characteristically simple group (?) that is non-abelian, then the automorphism group of $G$ is a Complete group (?).

In particular, this shows that the automorphism group of any Simple non-abelian group (?) is a complete group.

Related facts

Breakdown of analogue for abelian case

• Holomorph of elementary abelian group not implies every automorphism is inner

Monolith results for abelian and non-abelian cases

• Additive group of a field implies monolith in holomorph: In particular, any elementary abelian group is a monolith in its holomorph.
• Semidirect product with self-normalizing subgroup of automorphism group of coprime order implies every automorphism is inner
• Characteristically simple and non-abelian implies monolith in automorphism group

Other related facts

• Characteristically simple implies CSCFN-realizable
• Additive group of a field implies characteristic in holomorph
• Odd-order elementary abelian group is fully invariant in holomorph

Facts used

1. Center is characteristic
2. Centerless implies NSCFN in automorphism group
3. Characteristically simple and NSCFN implies monolith: If $G$ is a characteristically simple subgroup of a group $H$, such that $G$ is normal in $H$, $C_{H}(G)\leq G$, and every automorphism of $G$ extends to an inner automorphism of $H$, $G$ is a the unique nontrivial normal subgroup of $H$ contained in every nontrivial normal subgroup of $H$.
4. Monolith is characteristic
5. Centerless and characteristic in automorphism group implies automorphism group is complete

Proof

Given: A characteristically simple non-abelian group $G$, with automorphism group $\operatorname {Aut} (G)$.

To prove: $\operatorname {Aut} (G)$ is complete: it is centerless and every automorphism of it is inner.

Proof:

1. (Given data used: $G$ is characteristically simple non-abelian; Facts used: fact (1)): $G$ is centerless: By fact (1), the center of $G$ is characteristic in $G$. Since $G$ is non-abelian, the center of $G$ is not equal to $G$. Since $G$ is characteristically simple, this forces the center of $G$ to be trivial.
2. The natural mapping from $G$ to $\operatorname {Aut} (G)$ given by the conjugation action is injective, identifying $G$ with its image $\operatorname {Inn} (G)$: This follows from the fact that $G$ is centerless, so the kernel of the map is trivial.
3. (Facts used: fact (2)): $G$ is normal, self-centralizing and fully normalized in $\operatorname {Aut} (G)$: This follows from fact (2), and the fact that $G$ is centerless.
4. (Given data used: $G$ is characteristically simple; Facts used: fact (3)): By fact (3) and the conclusion of the previous step (step (3)), and the given datum that $G$ is characteristically simple, we conclude that $G$ is a monolith in $\operatorname {Aut} (G)$: it is the unique nontrivial normal subgroup of $\operatorname {Aut} (G)$ that is contained in every nontrivial normal subgroup of $\operatorname {Aut} (G)$.
5. (Facts used: fact (4)): $G$ is characteristic in $\operatorname {Aut} (G)$. This follows from step (3) and fact (4).
6. (Facts used: fact (5)): We showed that $G$ is centerless (step (1)) and characteristic in $\operatorname {Aut} (G)$ (step (4)). Thus, fact (5) yields the desired result.