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Consider the $\operatorname{Spin}^c$ group, i.e. the elements of norm $1$ in the complexified Clifford algebra $\mathbb{C}l^0$. Please tell me why this is isomorphic to $(SO(n) \times S^1)/\{1,-1\}$ where we identify $-1$ with $I \times (-1)$ for $I$ being the identity matrix in $SO(n)$.

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    A good place to find this type of thing is usually on the nLab. In particular you can find more info on [this](http://ncatlab.org/nlab/show/spin%5Ec) page2012-01-20

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It isn't. Rather, $\text{Spin}^c(n)$ is isomorphic to $\text{Spin}(n)\times_{\pm1} U(1)$. Perhaps you are thinking of the fact that $\text{Spin}^c (n) / \pm 1$ is isomorphic to $SO(n)\times U(1)$? (See section 2.6 in J. Morgan's book on the Seiberg-Witten equations.)