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Given that $ab=e$, we know $b =a^{-1}$. Since $G = \langle a,b\rangle$ this implies $\langle a,b\rangle=\langle a,a^{-1}\rangle=\langle a\rangle=G$ (as elements that generate the group). The presentation rewritten in terms of $a$ is trivial, i.e. $\langle a,a^{-1}\mid aa^{-1}=e\rangle$, so $G$ is a free group on the generator $a$, which is isomorphic to $\mathbb{Z}$.

I have just learned about group presentations and I am also wondering if there is a general method for identifying the isomorphism classes of a group given its presentation.

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  • $\begingroup$ No there is no general method, because most questions that you might ask, such as "is this group finite?" have been proved to be undecidable. $\endgroup$ – Derek Holt May 5 '16 at 19:28
  • $\begingroup$ Does this mean the conclusion of the proof is wrong $\endgroup$ – user52969 May 5 '16 at 19:41
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    $\begingroup$ No, not at all - it is correct. There are lots of methods that can be applied to particular presentations. I was just saying that there is no uinform method that can be applied to all presentations. $\endgroup$ – Derek Holt May 5 '16 at 19:46

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