1. Suppose that S is a semigroup, and that ax=b and ya=b has a solution for every a and b in S, prove that S is a group.

My proof:

Since ax=b and ya=b has a solution for every a and b, then ax=a and ya=a has a solution, therefore they are the left and right identity, implies that the identity exist in S. [Question: Do I need to show that y = x?]

Now, for inverse I'm not so sure if I can cancel any elements by multiplying y^-1 or x^-1, am I allow to do that?

2. Suppose that S is a finite semigroup such that the cancellation law holds, prove that it is a group.

Suppose that x,y,z are elements of S such that zx = zy, then x = y.

then $\displaystyle z^{-1}zx=z^{-1}zy$, now, am I allow to cancel any elements here? I guess I can't since I haven't find the inverses yet...

And why does it have to be finite? What if it is infinite semigroup?

Thank you very much for the helps!