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Let$G$ is a group, $a$ and $b$ are non-unit elements of $G$, $ab=bba$. …

Let $G$ be a group and $a,b\in G$ such that $ |a|=3, ab=b^2a, b\neq e. $

What can I say about $|b|$?

What I get so far is something like $ ba^2=a^2b, ab^2=b^4a. $ I suspect that one can not determine $|b|$, but I'm not able to give a proof.

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    @Brian, well-spotted. It seems my answer here is just an elaboration on a hint given there. I've left a comment there, linking here. Goku, don't delete --- but it will probably be closed, soon.2012-11-06

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You're given $aba^{-1}=b^2$. Then $a^2ba^{-2}=a(aba^{-1})a^{-1}=ab^2a^{-1}=(aba^{-1})^2=(b^2)^2=b^4$ Can you see how to get $a^3ba^{-3}$? And, once you have that, can you make use of $|a|=3$?

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    It's partly about experience, and partly about just trying stuff until you find something that works. You're told something about powers of $a$, you're given$a$relation between $a$ and $b$, you're asked something about powers of $b$: so you write stuff down and try various combinations of things until you come up with something that works. The point is not to be afraid to try lots of different things --- the worst thing that can happen is you try something that doesn't work, so you try something else. No bridges fall down, just because you tried something that didn't work.2012-11-06