# Thread: Prove quotient group cyclic

1. ## Prove quotient group cyclic

Let $G=C_8\times C_8$ and $H\leq G$. $H$ is cyclic of order 8.

Show that $G/H$ is cyclic.

2. ## Re: Prove quotient group cyclic

I'm not sure my solution is correct, but here it goes.

I assume $C_8$ is a cyclic group of order 8. So H is isomorphic $Z_8$, or to $\{0\} \times Z_8$. So $(C_8 \times C_8)/H$ is isomorphic to $(Z_8 \times Z_8)/(\{0\} \times Z_8)$, which is isomorphic to $(Z_8/ \{0\}) \times (Z_8/Z_8)$, which is isomorphic to $Z_8$, which is a cyclic group.

Edited to, hopefuly, correct the mistakes.

3. ## Re: Prove quotient group cyclic

Thanks, You are correct. We have atheorem that states $(H\times K)/H \cong K$.