Suppose we have a finitely presented group $G$ with free presentation
$$
R\hookrightarrow F \twoheadrightarrow G.
$$
To this presentation we may associate an extension with abelian kernel
$$
R/R' \hookrightarrow F/R' \twoheadrightarrow G.
$$
Here $R'$ denotes the commutator subgroup of $R$, and the $G$-module $R/R'$ is sometimes known as the *relation module* associated to the presentation.



> 
> Under what conditions is the group $F/R'$ finitely presentable?
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To give a concrete example, consider the free product of cyclic groups $C\_2\ast C\_3$ with presentation $\langle a,b \mid a^2,b^3 \rangle$. 



> 
> Is there a nice finite presentation of $F/R'$ in this case?
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I would also be interested in any pointers to places in the literature where calculations for particular presentations are carried out. 



Calculations with relation modules
Let $F$ be a finitely generated non-abelian free group with a non-trivial normal subgroup $R \lhd F$. Suppose that $G:=F/R$ is finitely presented. Then the group $F/R'$ is finitely presented if and only if $G$ is finite. In particular, if $G=C\_2\*C\_3$ then $F/R'$ is not finitely presented.


This is mentioned in the paper of Kulikova and Ol'shanskii (MR2523056
Kulikova, O. V.; Olʹshanskiĭ, A. Yu.
"On the finite presentability of the group $F/[M,N]$." (Russian. Russian summary) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2006, no. 6, 19--21, 62; translation in
Moscow Univ. Math. Bull. 61 (2006), no. 6, 20–23 (2007); <http://arxiv.org/abs/math/0703604>) and is attributed to Shmel'kin (MR0193131 Šmelʹkin, A. L. "Wreath products and varieties of groups." (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 29 1965 149–170).


