Find a group that the additive group $\mathbb{R} / \mathbb{Z}$ is isomorphic to

Question

Find a group that the additive group $\mathbb{R} / \mathbb{Z}$ is isomorphic to

What I understand is:

$\mathbb{R} / \mathbb{Z} = \{ a + \mathbb{Z} \,| \, a \in \mathbb{R}\}$ and the kernel is $0+ \mathbb{Z} = \mathbb{Z}$

I'm still a bit unclear on factor groups

Answer 1

You can consider as it is said in comments the group $[0,1)$ with the operation $x\oplus y=x+y-[x+y]$. To prove that this group is isomorphic to $\mathbb{R}/\mathbb{Z}$ you have to recall the first isomorphism theorem and find a homomorphism $\phi: \mathbb{R}\rightarrow [0,1)$ such that the $\text{ker }\phi=\mathbb{Z}$. Try $\phi(x)=x-[x]$ and then conclude that $\mathbb{Z}$ is isomorphic to $[0,1)$. Can you filled the details?

Answer 2

It is isomorphic it to circle group.

$\phi: \mathbb{R} \to U$

$\phi(r + \mathbb{Z}) = e^{i(2\pi r)}$, The $kernel(\phi)$ is everything that maps to 1.

it is surjective, a homomorphism and using the first isomorphism theorem

if $\phi$ is surjective then $\mathbb{R}/ker(\phi) \simeq U$ and $ker(\phi) = \mathbb{Z}$ so $\mathbb{R}/ \mathbb{Z} \simeq U$