Consider the group G=\mathbb{Z}_2\ast\mathbb{Z}_3=<x,y:x^2=y^3=e> and consider the surjective homomorphisms G\twoheadrightarrow\mathbb{Z}_2, G\twoheadrightarrow\mathbb{Z}_3 and G\twoheadrightarrow\mathbb{Z}_6. Find a presentation and the abelianization of each kernel.