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    Unitary

    Is U(1) isomorphic to interet mod n?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    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}
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