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How do I prove that if $a$, $b$ are elements of group, then $o(ab) = o(ba)$?

For some reason I end up doing the proof for abelian(ness?), i.e., I assume that the order of $ab$ is $2$ and do the steps that lead me to conclude that $ab=ba$, so the orders must be the same. Is that the right way to do it?

  • 4
    Why on earth are you assuming that the order of $ab$ is $2$?2012-12-08
  • 0
    This question is related to http://math.stackexchange.com/questions/225942/a-b-in-g-has-finite-order-then-is-the-order-of-ab-ba-a-1b-1-with2013-04-22
  • 0
    See [this question](http://math.stackexchange.com/questions/1086198/deduce-lvert-ab-rvert-lvert-ba-rvert-for-all-a-b-in-g-where-g-is-a-gro/1086267#1086267) for answers that go more to the essence of the matter (conjugation).2016-12-20

6 Answers 6