Let $\phi: G\to H$ be a group homomorphism. Let $S\subset G$ be a set of generators of $G$, then $\phi$ is an epimorphism iff the set $\phi(S)$ generates $H$.
I wrote the definition of surjective map: $\forall h\in H \exists g\in G:\phi(g)=h$.
But how to proceed from here?
What I thought is that $\phi$ is onto iff $\phi(G)=H$ iff $H=\phi(G)=\phi(\langle S \rangle)=\langle \phi(S) \rangle $
But how to show the last equality?
Best Answer
Hint The subgroup generated by $S$ has an explicit description given by $\{s_1^{k_1}\cdot\dots\cdot s_n^{k_n}\mid n\in \Bbb N, s_1,…,s_n\in S, k_1,…,k_n\in \Bbb Z\}$. Use this and the defining property of a group homomorphism to obtain the last equality.