TITLE: How to test whether two group presentations are isomorphic
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: 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.