I've been proposed by a very interesting theoretical problem in my abstract algebra class. It seems like one of those open problems with a concise and simple wording, but the proof and answer could probably take up more than 20 pages to explain.
Given any set (nonempty and finite), is it possible to impose a binary operation on the set such that it can turn into
1) A group
2) An abelian group
3) A cyclic group
I think answering (3) will answer (2) immediately. Anyways, I thought about it and I think for (3), I just have to take an element, keep multiplying by itself to create a cyclic.
As for (1), I am not sure. I think if you have some ingenuity, you could. But there could be some crazy sets out there that doesn't have this property
EDIT: Okay, so the bijective map and the inverse part I got. So I take an element from the set S and map it to T. That is I get $\phi(a * b) = \phi(a) *' \phi(b)$.
I don't understand how you got the + operation and I don't understand what you mean by "evaluation by last residues"