Could $\langle \Gamma | R \rangle \cong \langle \Gamma | S\rangle$ if $\langle R\rangle \subsetneq \langle S\rangle$?

Question

If we have two finitely presented groups $\langle \Gamma | R\rangle$ and $\langle \Gamma | S\rangle$ with $\langle R\rangle \subsetneq \langle S\rangle$, could they be isomorphic?

Answer 1

Yes. An example is $\langle x,y \mid x \rangle$ and $\langle x,y \mid x,y^{-1}xy \rangle$.

Both presentations define an infinite cyclic group, but $\langle x \rangle$ and $\langle x,y^{-1}xy \rangle$ are distinct subgroups of the free group on $x,y$. The first is cyclic and the second is free of rank $2$ on its generators.