The statement of the Frucht's Theorem as follows: "Every Finite Group is Automorphism Group of some graph." The proof involves a result that the group of color preserving Automorphisms of a Cayley Color Graph of a finite group is isomorphic to the group itself.
I am confused how to use the condition of 'finite group' in the proof of above results. I suspect that the second result is true for all groups and not necessarily for finite groups. And in that case Frucht's Theorem should be true for all Groups. Where I am going wrong?
It's true for every group, Frucht was only able to prove it for finite groups. The proof for infinite groups is actually a lot harder.
You can find it right here: https://link.springer.com/article/10.1007/BF01319053
(you can find it on sci hub)