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Suppose I have two presentations for groups:

$\langle x,y|x^{7} = y^{3} = 1, yx = x^2y\rangle$ and $\langle x,y|x^{7} = y^{3} = 1, yx = x^4y\rangle$

What is the standard approach to deciding whether the presentations are isomorphic?

I'm working through an application of Sylow Theory which classifies groups of order $21$.

In the text it says that these two presentations above are isomorphic, but I cannot see how to prove it or even suspect it.

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Well, you'd want to find a set of generators of the first group that satisfied the relations of the second group. If we rewrite $yx=x^2y$ as $yxy^{-1}=x^2$ (which turns this somewhat abstract equality into something a bit more concrete), we see immediately that $y^2xy^{-2}=x^4$ and indeed since $y^2$ is of order 3, $x,y^2$ are the generators you're looking for.

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    That was the intended implication but the argument is certainly clearer (and equally short) if I say $y^2$ is of order 3, so I have edited my answer accordingly.2012-10-12