Is there difference between finitely presented groups and finitely generated groups? A group is said to be finitely generated if it can be generated by a finite set of generators. 
Question : Is there difference between finitely presented groups and finitely generated groups?

Here is a sketch of proof:


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*$\langle a,t \mid [t^nat^{-n},t^{m}at^{-m} ]=1, \ n,m \in \mathbb{Z} \rangle$ is a presentation of $L:= \mathbb{Z} \wr \mathbb{Z}$.

*If $L$ is finitely presented, there exists a finite set $S \subset \mathbb{Z}$ such that $$\langle a,t \mid [t^nat^{-n},t^{m}at^{-m}]=1, \ n,m \in S \rangle$$ is isomorphic to $L$, so it is sufficient to show that $$L_k= \langle a,t \mid [t^nat^{-n},t^{m}at^{-m}]=1, -k \leq n,m \leq k \rangle$$ is not isomorphic to $L$.

*Setting $a_n=t^nat^{-n}$, we get $$L_k= \langle a_{-k}, \dots, a_k,t \mid [a_n,a_{m}]=1, a_{n+1}=ta_nt^{-1}, -k \leq n,m \leq k \rangle.$$

*Notice that $L_k$ is a HNN extension and use Britton lemma to show that the subgroup $\langle a_0,t \rangle$ is isomorphic to the free product $\mathbb{Z} \ast \mathbb{Z}$.

*Therefore, $L_k$ is not solvable so it cannot be isomorphic to $L$ (which is solvable).
