I was told that $\mathbb{R}$$/$$\mathbb{Z}$ is isomorphic to $\mathbb{R}/2\pi \mathbb{Z}$ when these groups are taken under addition. Is this always true? I do not specifically see why this has to be true.
Thoughts: Use the exponential map and then the kernel is all multiples of $2\pi$, then use the first isomorphism theorem?
On
For any nonzero real number $c$, there is an isomorphism of the additive group $(\mathbb R,+)$ with itself given by $x\mapsto c x$. In our case, we want to take $c=2\pi$.
Under this isomorphism $\mathbb Z$ maps to $2\pi \mathbb Z$.
Because of this, the two quotients $\mathbb R/\mathbb Z$ and $\mathbb R / 2\pi \mathbb Z$ are definitely isomorphic, and in fact the corresponding map on the quotients $t+\mathbb Z \mapsto 2\pi t + 2\pi \mathbb Z$ is an isomorphism.
The following map is an isomorphism between these. $$x + \mathbb Z \mapsto 2\pi x + 2\pi \mathbb Z$$