Characteristically simple and non-abelian implies automorphism group is complete

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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

Monolith results for abelian and non-abelian cases

Other related facts

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) \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. 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.