Although the following question is not phrased in the most accurate way, I would like to ask it in the same way it rushed to my mind:
"Looking at some basic examples of group theory with geometrical interpretations, such as the group of symmetries of a regular polygon ${p}$ or a regular polyhedron ${p,q}$, where the notion of Duality makes geometric sense, and using the fact that every group would be completely identified by a presentation of the form $G=<S,R>$, where $S$ and $R$ are respectively the set of generators and relations, I was wondering if there is an algebraic notion of duality of a group $G=<S,R>$, in the sense that simply the set of relations $R$ and generators $S$ swap their roles, for the dual group we obtain. Namely, could we construct a new group with presentation of the form $G^D=<R,S>$ ?"
It seems quite feasible to take the set of relations of one group as the set of generators of another. However, when it comes to the set of generators of the first group, I don't see any clear approach to use them as the set of relations!
I assume, supposed $G=\langle S\mid R\rangle$, you somehow want to interpret $S$ as the corresponding set of some relations, using elements of $R$ as variable symbols.
Well, such an interpretation cannot exist, at least in its full generality, simply because $R$ can be empty without $S$ being empty (representing the free group over $S$), but if $S$ is empty, then we don't have any variable symbol to write up any relation, so $R$ also has to be empty.