Proof of Frucht's theorem

Proof

The main idea of the proof is to observe that the Cayley graph of G, with the addition of colors and orientations on its edges to distinguish the generators of G from each other, has the desired automorphism group. Therefore, if each of these edges is replaced by an appropriate subgraph, such that each replacement subgraph is itself asymmetric and two replacements are isomorphic if and only if they replace edges of the same color, then the undirected graph created by performing these replacements will also have G as its symmetry group.