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Problem (Conjugacy Is An Equivalence Relation)
Show that conjugacy is an equivalence relation on $G$. In other words, show the following three things.
- For each $a\in G$, we know that $a$ is a conjugate of $a$.
- If $a$ is a conjugate of $b$, show that $b$ is a conjugate of $a$.
- If $a$ is a conjugate of $b$ and $b$ is a conjugate of $c$, show that $a$ is a conjugate of $c$.
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