homomorphism and isomorphism?

Question

Let G=(R,+) and let H be an orthogonal group.
Show that the function f:G->H given by f:(theta)->rot(theta) is a homomorphism.
Is it an isomorphism?
What about g:(theta)->ref(theta)?

CR Jordan and DA Jordan pg 104

tanx a bunch!

Answer 1

You haven't defined "rot" and "ref", but if they have their usual meaning as rotation and reflection then there are plenty of non-zero thetas that get mapped to the identity element.

So f and g wouldn't be isomorphisms.