How to show $\frac{\langle x,y\rangle }{H}\cong\mathbb{Z}\oplus \frac{\mathbb{Z}}{2}$?

Question

Suppose an abelian group $K$ is generated by two elements $\langle x,y\rangle$ and $H$ is a subgroup. Now suppose $nx + my \in H \iff n=0$ and $m$ is even. Then $\frac{\langle x,y\rangle}{H}\cong\mathbb{Z}\oplus \frac{\mathbb{Z}}{2}$. How do I properly show this? Is it a consequence of the fundamental theorem of finitely generated groups?

Answer 1

Since $H \subset K = \langle x, y \rangle$ is a subgroup, a generic element of $H$ will be written as $nx + my$. Then $nx + my \in H \iff n=0$ and $m$ is even is equivalent to saying that elements of $H$ are of the form $2ky$ where $k$ is an integer. Then $H = \langle 2y \rangle$ so $$ \frac{\langle x, y \rangle}{H} = \frac{\langle x, y \rangle}{\langle 2 y \rangle} \cong \langle x \rangle \oplus \frac{\langle y \rangle}{\langle 2y \rangle} \cong \mathbb{Z} \oplus \mathbb{Z}/2. $$