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Let $G$ be a compact matrix group. May I know why the conjugacy classes of $G$ is necessarily closed? I tried to argue by taking limits but to no avail so is there a hint on how to tackle this problem?

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Let $G$ be a matrix group, and suppose a sequence $g_i^{-1} x g_i$ of elements in the conjugacy class of $x$, converges to a point $y \in G$. By compactness, the sequence $\{g_i\}$ has a convergent subsequence with a limit $g$. Therefore $y = g^{-1} x g$, and hence $y$ is also in the conjugacy class of $x$. Hence the conjugacy class of $x$ is closed.

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