# Characteristically simple and non-abelian implies automorphism group is complete

From Groupprops

## Contents

## Statement

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

- 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

- 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

- Center is characteristic
- Centerless implies NSCFN in automorphism group
- Characteristically simple and NSCFN implies monolith: If is a characteristically simple subgroup of a group , such that is normal in , , and every automorphism of extends to an inner automorphism of , is a the unique nontrivial normal subgroup of contained in every nontrivial normal subgroup of .
- Monolith is characteristic
- Centerless and characteristic in automorphism group implies automorphism group is complete

## Proof

**Given**: A characteristically simple non-abelian group , with automorphism group .

**To prove**: is complete: it is centerless and every automorphism of it is inner.

**Proof**:

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