# Proving a group $G$ with presentation $\langle a,b\mid ab\rangle$ is isomorphic to $\Bbb Z$.

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.

• 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. – Derek Holt May 5 '16 at 19:28
• Does this mean the conclusion of the proof is wrong – user52969 May 5 '16 at 19:41
• 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. – Derek Holt May 5 '16 at 19:46