Groups with trivial automorphism group?

Hi Josh, here's my proof (which uses AC, sorry :)). Suppose to the contrary that |G| > 2. Take distinct elements g, h in G which are different from the identity. Since conjugation by an element of G is an automorphism of G and Aut(G) = 1, G is abelian. Then inversion in G is an automorphism of G, so G has exponent two. Hence G is an F2 -vector space. Let S be a basis for G containing g and h. Define a linear transformation T : G -> G such that T(g) = h, T(h) = g, and T(v) = v for all v in S - {g,h}. It belongs to Aut(G), but it differs from the identity automorphism. This is a contradiction.