Sorry if this is an evident question but I don't know how to start answering it. If the group $G$ is abelian then both elements $hg$ and $gh$ are the same, so its order is evidently equal. But if commutativity is not a given, how can one state that two different elements would always have the same order? Is there some "ordinary" group property that I am missing? Or should another property about $G$ be stated (finite, normal, state precisely its generators etc.)? Any insight would be very much appreciated.
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https://math.stackexchange.com/questions/3741886/how-do-two-conjugate-elements-of-a-group-have-the-same-order – Randall Nov 29 '21 at 19:15
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Indeed, $gh$ and $hg$ are conjugate in $G$, so they have the same order. This has been proved here already in several duplicates. – Dietrich Burde Nov 29 '21 at 19:24
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To see that they are conjugate, note that $(ghg)^{-1}gh(ghg)=hg$ – Randall Nov 29 '21 at 19:44
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1@Randall Or just $g^{-1}(gh)g=hg$. – Servaes Nov 29 '21 at 20:40
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Yikes, right. Much simpler. – Randall Nov 29 '21 at 21:00