What does operation mean? How many mappings are there? I think once again it would be easier to show combinatorially that2) How many operations are there on S?
I know this is a little above what you are doing right now, but this will look EXACTLY like . But, for now draw up the Cayley and fill in the squares remembering that etc.
3) Find the Caley table for an operation such that is a group with identity element a.
You can do the rest.