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 $$ \frac{\langle x, y \rangle}{H} = \frac{\langle x, y \rangle}{\langle 2 y \rangle} \cong \langle x \rangle \oplus \frac{\langle y \rangle}{\langle 2y \rangle} \cong \mathbb{Z} \oplus \mathbb{Z}/2. $$

  • $\begingroup$ 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 $\frac{G}{H_1\oplus H_2} \cong \frac{G_1}{H_1}\oplus \frac{G_2}{H_2}$ $\endgroup$
    – roi_saumon
    Jun 26 '19 at 8:57
  • $\begingroup$ 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. $\endgroup$
    – mathphys
    Jun 26 '19 at 12:08

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