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Let G be the group defined by the following generators and relations: $G = \left\langle a, b, c \mid ab = ba, ac = ca, bc = cba, a^{3} = 1, b^{3} = 1, c^{3} = 1\right\rangle.$ Show that $\langle a\rangle$ is normal in $G$ and $|G| = 27.$

I know how to do the first part, but I am not sure the second part. I am trying to show that $\bigl\lvert\langle b ,c\rangle\bigr\rvert = 9$, it is the right way?

Cameron Buie
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    I think the easiest way is to use the relations to show that any element may be written on the form $c^ib^ja^k$. – Arthur Dec 27 '15 at 15:21
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    I would do a "Word computation". Show (with the help of your relations) that every word/element can be written as $a^n c^k b^l$. Then use the last 3 relations to conclude that there are 27 possibilities – M.U. Dec 27 '15 at 15:25
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    I think your idea is spot on, and it is a fairly easy thing to do. – Tobias Kildetoft Dec 27 '15 at 15:44

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