# Math Help - Unitary

1. ## Unitary

Is U(1) isomorphic to interet mod n?

2. Originally Posted by alexandrabel90
Is U(1) isomorphic to interet mod n?
I assume that $U(1)$ is the unitary group of one by one matrices? Clearly (not to insult your intelligence) $\phi:U(1)\to\mathbb{T}:[z]\mapsto z$ is an isomorphism (where $\mathbb{T}$ is the circle group) but it's also clear that $\mathbb{T}\cong\mathbb{R}/\mathbb{Z}$ since the map $\phi:\mathbb{R}\to\mathbb{T}:r\mapsto e^{2\pi i r}$ is a homomorphism with kernel $\mathbb{Z}$ (just use the FIT). This is sort of close to what you want? It's also true that $U(1)\cong\text{SO}(2)$ (the special orthogonal group) where I'm sure it's clear to you that you just send $\mathbb{T}\ni a+bi\mapsto \begin{pmatrix}a & b\\ -b &a\end{pmatrix}$.

So, the answer to your question is no. Is there a particular reason you think that? You know that $\#(U(1))\geqslant\infty$ since $[e^{ir}]$ is an element for $r\in\mathbb{R}$