Trivial automorphism group implies trivial or order two

Statement
If the automorphism group of a group is the trivial group, then the group is either itself trivial, or cyclic of order two.

Related facts

 * Coprime automorphism group implies cyclic with order a cyclicity-forcing number

Proof outline
The proof has three steps:


 * The group must be Abelian: Otherwise, it would have nontrivial inner automorphisms
 * The group must have exponent two: Otherwise, the map sending an element to its inverse, would be a nontrivial automorphism. In particular, it is an elementary Abelian group, or is a vector space
 * The group must be one-dimensional as a vector space, otherwise it will have other automorphisms (the general linear group on a vector space of dimension greater than one, is nontrivial)