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The relation $gh = hg$ means this group $G$ is commutative. $\langle g\rangle$ and $\langle h \rangle$ are cyclic subgroups of G. Still have no idea how to conclude $|gh|$ is finite.

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    HINT: If $gh=hg$, then can you rewrite $(gh)^n$ as a product of a power of $g$ and a power of $h$?2012-02-13
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    Can you see what's the value of a power $(gh)^n$?2012-02-13
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    **Note:** The condition $gh=hg$ does **not** mean that the group $G$ is commutative. It only means that $g$ and $h$ commute. It is perfectly possible for two elements in a noncommutative group to commute.2012-02-13

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