Quotient of a finitely presented group by a finitely presented group

Question

Is a quotient of a finitely presentable group $G$ by a subgroup $N$ necessarily finitely presentable? What about if the subgroup $N$ is also finitely presentable?

Answer 1

If $N$ is finitely generated, then you can just add the generators to the relation set of $G$ to get $G/N$, which will be a finite presentation. (after all a presentation is just a list of generators and things which normally generate some normal subgroup).

And Mariano's answer is right, although it isn't completely trivial to construct/show a group is infinitely presented, but finitely generated. Some well known groups are $B \wr A$ (restricted wreath product) where $B$ is nontrivial and $A$ is infinite. Another way to construct infinitely presented groups is through small cancellation theory.

Answer 2

Since there exist finitely generated groups without finite presentations, the answer to your first question is clearly no, since free groups of finite rank are finitely presentable.