Show $\phi$ is a homomorphism.

Question

Let $\phi :\mathbb{R}\rightarrow \mathbb{S}$ be defined by

$\hspace{150pt} \phi(r)=e^{2\pi ri}=\cos(2\pi ri)+i\sin(2\pi ri)$

Show that this is a group homomorphism.

$\mathbb{S}$ is the unit circle.

---

If I consider two elements $q,r\in\mathbb{R}$ then

$\hspace{100pt} \phi(qr)=e^{2\pi(qr)i}=\cos(2\pi (qr)i)+i\sin(2\pi (qr)i)$

and

$\hspace{50pt} \phi(q)\phi(r)=e^{2\pi qi}e^{2\pi ri}=(\cos(2\pi qi)+i\sin(2\pi qi))(\cos(2\pi ri)+i\sin(2\pi ri))$

I haven't been able to make $\phi(qr)=\phi(q)\phi(r)$ though. Am I missing an identity or something?

Answer 1

Remember that groups have operations with associative laws. You did not note that $\mathbb{R}$ is the additive group of reals. Now you can take $a,b\in \mathbb{R}$ with $a+b \in \mathbb{R}$ since groups are closed. Then $\phi(a+b)=e^{(a+b)(2\pi i)}=e^{2\pi ai}e^{2\pi bi}=\phi(a)\phi(b)$ and so $\phi$ is a homomorphism.

Answer 2

There's no need to use cosine and sine simply note: $e^{2\pi qi}e^{2\pi r i}=e^{2\pi i(q+r)}$.