Call a group presentation $\langle X \,\|\,R \rangle$ *minimal* if no relator from $R$ is a consequence of the remaining relators, i.e., no $r \in R$ belongs to the normal closure of $R\setminus \{r\}$ in the free group $F(X)$.


**Question:** Does every finitely generated group have a minimal presentation (with $X$ finite)?


*Remark:* evidently every finite presentation $\langle X \,\|\,R \rangle$ has a minimal sub-presentation $\langle X \,\|\,R' \rangle$ (where $R' \subseteq R$ has the same normal closure in $F(X)$ as $R$), so the question really concerns infinitely presented groups. 


It is possible to construct infinite presentations which do not have minimal sub-presentations. Indeed, let $F=F(a,b)$ be the free group on $\{a,b\}$. One can choose "sufficiently independent" elements $w\_1,w\_2,\dots \in F$ so that the presentation $$\langle a,b \,\|\, w\_i^2,w^2\_{i+1}w\_i, ~i \in \mathbb{N} \rangle$$ has no minimal sub-presentation. Here $R$ consists of the elements $w\_1^2,w\_2^2,\dots$, and $w\_2^2w\_1,w\_3^2w\_2,\dots$.


However, the above group also has the presentation $\langle a,b \,\|\,w\_1,w\_2,\dots \rangle$, which may be minimal.



Does every f.g. group have a minimal presentation?
The answer is **no**. 


In order to see this, you may combine Theorem 3.9 and Remark 5.3 of [1]. A counter-example is given by the nilpotent-by-Abelian group $B$ of Equation (3.2). Further examples are provided by Remark 5.15.




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[1] R. Bieri, Y. de Cornulier, L. Guyot and R. Strebel, ["Infinite presentability of groups and condensation"](http://www.ams.org/mathscinet-getitem?mr=3249690), 2014. 


