Isomorphism between group presentations [closed]

Question

Let $$G_1=\langle a,b,c:\hspace{0.2cm}abc^{-1}a^{-1}b^{-1}c=1\rangle,$$ $$G_2=\langle d,e:\hspace{0.2cm}de d^{-1}e^{-1}=1\rangle.$$ How can I construct (if possible) a group isomorphism between $G_1$ and $G_2$?

I try to think of this as identifying the edges of a square and a hexagon both representing the torus, but I get stuck.

Thanks for the comments and help.

Answer 1

These groups are not isomorphic because the abelianization of the first group is $\Bbb Z^3$ and the abelianization of the second group is $\Bbb Z^2$. In fact the first group is the free product $\Bbb Z*\Bbb Z^2=\langle b, ab, c^{-1}b\rangle$, where $(ab)$ commutes with $(c^{-1}b)$.