# Prove quotient group cyclic

• Sep 9th 2011, 11:26 AM
skyking
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.
• Sep 10th 2011, 02:17 AM
ModusPonens
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.
• Sep 10th 2011, 05:54 AM
skyking
Re: Prove quotient group cyclic
Thanks, You are correct. We have atheorem that states $(H\times K)/H \cong K$.