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Is there a way to find the group whose presentation is given by $\langle a, b, c \mid a^{2} = b^{2} = c^{2} = abc = e, ab = ba, ac = ca, bc= cb\rangle$?

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    Since ab=ba, ac=ca, and bc=cb, the group is abelian. Since aa=bb=cc=e, it is generated by a bunch of (ok just three) elements of order 2. Check the abelian groups you know that are generated by elements of order 2. It should be the second one you try.2011-11-03
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    $ab=c^{-1}=c$ (since $c^2=e$ so $c=c^{-1}$. Looks like the Klein 4-group.2011-11-03
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    Perhaps interesting to note that you don't even need the "commuting" relations; $a^2=b^2=c^2=abc=e$ is already enough to deduce the structure of the group.2011-11-03

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