Let $G = \mathbb{Z}_{512} \times \mathbb{Z}_{1729}$ and let $H = <([0],[1])>$. Show that $G/H$ is isomorphic to $\mathbb{Z}_{512}$.

Question

Let $G = \mathbb{Z}_{512} \times \mathbb{Z}_{1729}$ and let $H$ be the subgroup generated by $([0],[1])$. Show that $G/H$ is isomorphic to $\mathbb{Z}_{512}$.

This is using direct products.

First I tried to get $H$

$H = \{([0],[1])\}$ Since $\forall [a] \in \mathbb{Z}_n [a][0] = [a0]=[0]$ and $[a][1] = [a]$, so $([0],[1])^n = ([0],[1]) \forall n \in \mathbb{N}$

$G/H = \{g_1H, g_2H,... \} = \{g_1\{([0],[1])\}, g_2\{([0],[1])\}, ...\}$

For an arbitrary $([a],[b])$ then $([a],[b])([0],[1])=([0],[b])$

Then $G/H = \{\{([0],[b])\} : [b] \in \mathbb{Z}_{1729}\}$

Define $\phi:G/H \to \mathbb{Z}_{512} : \phi(\{[0],[b])\}) = [b]$

I am stuck. What I am seeing is that it is isomorphic to $Z_{1729}$. What am I doing wrong or how can I "truncate" the $[b]$ values to $\mathbb{Z}_{512}$?

Answer 1

Hint: Prove that the map $\mathbb{Z}_{512} \times \mathbb{Z}_{1729} \to \mathbb{Z}_{512}$ given by $(x,y) \mapsto x$ is surjective. Find its kernel.