In Hungerford's "Algebra", he states Von Dyck's theorem as follows: Let X be a set and Y a set of reduced words on X, and G the group defined by generators $x\in X$ and relations $w=e, w\in Y$. If $H=\langle X\rangle$ and H satisfies all the relations in Y, then there exists and epimorphism $G\rightarrow H$.
But I have trouble with $H=\langle X\rangle$. If $G=\langle X\rangle$ and $H=\langle X\rangle$, then isn't it the same group? What am I missing here?
Thank you.