TITLE: What's a group whose group of automorphisms is non-abelian?
QUESTION [31 upvotes]: I recently attended an interview for admission to graduate programs in Mathematics. The interviewing professor asked me a question - Tell me a group whose group of automorphisms is non-abelian.
Because I was too nervous, I couldn't think of anything substantial.
Could someone please tell me, in such a situation, what is the logical way of deducing the answer to such a question posed by the interviewer?
Thank you very much for your help!

REPLY [3 votes]: $\DeclareMathOperator{Aut}{\operatorname{Aut}}$$\Aut(\Bbb C,+)$ contains the conjugation mapping $z \mapsto z^*$ and the multiplication by $i$ mapping as automorphisms. Now notice that $(iz)^* = -iz^* \neq iz^*$ for $z \neq 0$.