# Finite Presentation of a subgroup

I have the group $$\langle a,b \mid a^3b^3\rangle$$ Now I send both $$a$$ and $$b$$ to the generator of $$\mathbb{Z}/3\mathbb{Z}$$. This gives a well-defined homomorphism from our group to $$\mathbb{Z}/3\mathbb{Z}$$ and I am asked to find a finite presentation of the kernel of this homomorphism. How do I generally tackle these kind of questions?

I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far.

• You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations. – Derek Holt Jan 5 at 14:16
• I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far. – Jack Copper Jan 5 at 19:47
• Ah, that's a different method, and I will leave someone else to help you with that. – Derek Holt Jan 5 at 20:18
• Hint: Start with the wedge of two circles $X=S^1 \vee S^1$ and consider a homomorphism $h: \pi_1(X)\to Z/3$ sending each natural generator to the generator of $Z/3$. Can you identify the 3-fold cover of $X$ corresponding to the kernel of $h$? Nw, add some 2-cells... – Moishe Kohan Jan 5 at 23:03