Characteristically simple and non-abelian implies automorphism group is complete

Statement
If $$G$$ is a fact about::characteristically simple group that is non-abelian, then the automorphism group of $$G$$ is a fact about::complete group.

In particular, this shows that the automorphism group of any fact about::simple non-abelian group is a complete group.

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) uses::Center is characteristic
 * 2) uses::Centerless implies NSCFN in automorphism group
 * 3) uses::Characteristically simple and NSCFN implies monolith: If $$G$$ is a characteristically simple subgroup of a group $$H$$, such that $$G$$ is normal in $$H$$, $$C_H(G) \le G$$, 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) uses::Monolith is characteristic
 * 5) uses::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.