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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.

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    $\begingroup$ You could use the Reidemeister-Schreier algorithm. You need to provide more context about your knowledge of the theory of group presentations. $\endgroup$ – Derek Holt Jan 5 at 14:16
  • $\begingroup$ 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. $\endgroup$ – Jack Copper Jan 5 at 19:47
  • $\begingroup$ Ah, that's a different method, and I will leave someone else to help you with that. $\endgroup$ – Derek Holt Jan 5 at 20:18
  • $\begingroup$ 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... $\endgroup$ – Moishe Kohan Jan 5 at 23:03

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