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Math Help - Factor group order question

  1. #1
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    Factor group order question

    <x> \cong C_9 \cong <y> and G = <x> \times <y>.
    What is the factor group G/<x^3,y^3> order and write it as the direct product of cyclic groups.

    Isn't 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".

    Thank you for your help.
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  2. #2
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    Quote Originally Posted by kleenex View Post
    <x> \cong C_9 \cong <y> and G = <x> \times <y>.
    What is the factor group G/<x^3,y^3> order and write it as the direct product of cyclic groups.

    Isn't 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".

    Thank you for your help.
    Instead of \left< x^3,y^3\right> I think you mean \left<x^3 \right> \times \left< y^3 \right>. Then it means, 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.
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