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Math Help - Prove quotient group cyclic

  1. #1
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    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.
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  2. #2
    Member ModusPonens's Avatar
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    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.
    Last edited by ModusPonens; September 10th 2011 at 03:37 AM.
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  3. #3
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    Re: Prove quotient group cyclic

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