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

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?

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 $$\frac{\langle x, y \rangle}{H} = \frac{\langle x, y \rangle}{\langle 2 y \rangle} \cong \langle x \rangle \oplus \frac{\langle y \rangle}{\langle 2y \rangle} \cong \mathbb{Z} \oplus \mathbb{Z}/2.$$
• Thanks! But why $\frac{\langle x, y \rangle}{\langle 2 y \rangle} \cong \langle x \rangle \oplus \frac{\langle y \rangle}{\langle 2y \rangle}$? Did you use that $G=G_1\oplus G_2$ and $H_i$ subgroups of $G_i$ then $\frac{G}{H_1\oplus H_2} \cong \frac{G_1}{H_1}\oplus \frac{G_2}{H_2}$ Jun 26 '19 at 8:57
• You're welcome! Yes, this works; $K = \langle x, y \rangle \cong \langle x \rangle \oplus \langle y \rangle$. Also, $\langle 2y \rangle \cong 0 \oplus \langle 2y \rangle$ (here $0$ is the trivial group). Then applying the result you've mentioned above gives the result. Jun 26 '19 at 12:08