TITLE: Is there a non-trivial group G isomorphic to Aut(G)?
QUESTION [11 upvotes]: The title basically says it all.
Is there a group with more than one element that is isomorphic to the group of automorphisms of itself?
I'm mainly interested in the case for finite groups,
although the answer for infinite groups would still be somewhat interesting.

REPLY [3 votes]: See also Problem 2015-4C in the Nieuw Archief voor Wiskunde.
A solution (of mine) can be found here:
http://www.nieuwarchief.nl/home/problems/pdf/uitwerking-2015-4.pdf