I have a question that I have approached, but want to check if I'm on the right track.

Let G denote the group of symmetries of a circle. There are infinitely many reflections and rotations. There are no elements besides reflections and rotations. The identity element is the rotation by zero degrees. Let H denote the subgroup of H consisting of only the rotations.

Question: Define f: G--> {1,-1} by f(x) = 1 if x is a rotation and f(x) = -1 if x is a reflection. Is f a homomorphism?

What I did was check if f(xy)=f(x)f(y). So f((-1)*1)=f(-1)f(1)=-1, and f((1)*1)=f(1)f(1)=1, etc. It all checks out, so f is a homomorphism.

Did I do it right? Thanks!!