I define $\text{Ab}(G)=G/[G,G]$ where $[G,G]$ is the commutator subgroup. I want to show that $$\text{Ab}(G_1*G_2)\cong \text{Ab}(G_1)\oplus\text{Ab}(G_2)$$ This page gives a categorical proof, but I don’t know much category theory. Can someone give a purely group-theoretic proof of this (I know the universal property of abelianizations)? By the universal property, it would suffice to show that […]

I am a little confused about the definition of free products. Given a collection of groups $\{G_\alpha\}_\alpha$ in order to create their free product, I don’t understand what properties these $G_\alpha$ must have. I tell you three different circumstances that I met studying this definition. 1) In some forums and very brief notes I read […]

Prove that $PSL(2,\mathbb{Z})=C_2 \star C_3$. Now $C_2 \star C_3=\langle a,b\ |\ a^2, b^3 \rangle$ i.e. the free product. But how do I show that presentation of $PSL(2,\mathbb{Z})$ is this?

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