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.
If $gh = hg$ in a group and $|g|$ and $|h|$ are finite, is $|gh|$ finite too?
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abstract-algebra
group-theory
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2HINT: 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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0Can you see what's the value of a power $(gh)^n$? – 2012-02-13
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2**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