Which groups have trivial automorphism group?
Solution
The only groups with trivial automorphism group are \[ 1\qquad\text{and}\qquad C_2. \]
Suppose \(\operatorname{Aut}(G)=1\). Every inner automorphism is then trivial, so \[ \operatorname{Inn}(G)=1. \] Hence \[ G/Z(G)=1, \] and therefore \(G\) is abelian.
For an abelian group, inversion \[ x\longmapsto x^{-1} \] is an automorphism. Since every automorphism is trivial, inversion must be the identity, so \[ x=x^{-1} \] for every \(x\in G\). Thus every element has order dividing \(2\), and \(G\) is a vector space over \(\FF_2\).
If its dimension were at least \(2\), a nontrivial linear automorphism could interchange two basis vectors. Therefore the dimension is at most \(1\), giving \[ G=1\quad\text{or}\quad G\cong C_2. \]
Conversely, both \(1\) and \(C_2\) plainly have only the identity automorphism.