I know that in the quaternion group $Q_8$, there are $5$ conjugacy classes, $2$ of which are $\{1\}$ and $\{-1\}$. Why are $1$ and $-1$ in their own conjugacy classes rather than a single one: $\{1, -1\}$? Thanks!

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    Because they are both in the center. Observe that $g1g^{-1}=1$ for all $g\in G$ with any group $G$. Hence $1$ is always in a conjugacy class by itself. – Jyrki Lahtonen Apr 13 '17 at 19:52

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