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Consider the fundamental group $\pi_1(\mathbb{CP}^1\backslash \{a_1, \ldots, a_n\})$. It is said that there is a representation: $\pi_1(\mathbb{CP}^1\backslash \{a_1, \ldots, a_n\}) \to GL(n, \mathbb{C})$. I am confused with the order of these two groups. Since $\pi_1(\mathbb{CP}^1\backslash \{a_1, \ldots, a_n\}) \simeq \mathbb{Z}^{n-1}$, the order of this group is infinite. What is the order of the group $GL(n, \mathbb{C})$? Is the order of the group $GL(n, \mathbb{C})$ larger than the order of $\mathbb{Z}^{n-1}$?

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    It should be pointed out that a representation doesn't have to have the same order as the group. For example, the map g $\mapsto$ Id is a rep.2012-03-06

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Well, the cardinality of $\mathrm{GL}_n(\mathbb{C})$ is the same as that of the continuum $\mathbb{R}$, which is bigger than both that of $\mathbb{Z}^{n-1}$ and the fundamental group you are interested in (the free group on $n-1$ generators); these have the cardinality of $\mathbb{Z}$.

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    It's just something that I didn't fully realize right away when I was seeing this stuff for the first time. So now I yell it at people :) By the way, how do you italicize?2012-03-07