# Factor group order question

• Apr 9th 2008, 02:16 AM
kleenex
Factor group order question
$\displaystyle <x> \cong C_9 \cong <y>$ and $\displaystyle G = <x> \times <y>$.
What is the factor group $\displaystyle G/<x^3,y^3>$ order and write it as the direct product of cyclic groups.

Isn't $\displaystyle G/<x^3,y^3>$ has order 3(correct me if I'm wrong) and I need some help on this the second part "write it as the direct product of cyclic groups".

$\displaystyle <x> \cong C_9 \cong <y>$ and $\displaystyle G = <x> \times <y>$.
What is the factor group $\displaystyle G/<x^3,y^3>$ order and write it as the direct product of cyclic groups.
Isn't $\displaystyle G/<x^3,y^3>$ has order 3(correct me if I'm wrong) and I need some help on this the second part "write it as the direct product of cyclic groups".
Instead of $\displaystyle \left< x^3,y^3\right>$ I think you mean $\displaystyle \left<x^3 \right> \times \left< y^3 \right>$. Then it means, $\displaystyle G/\left<x^3 \right> \times \left< y^3 \right> = \left< x \right> \times \left< y\right> / \left<x^3 \right> \times \left< y^3 \right> \simeq \left< x \right> / \left<x^3 \right> \times \left< y\right> / \left< y^3 \right>\simeq \mathbb{Z}_6\times \mathbb{Z}_6$.