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Math Help - Prove that a quotient group is cyclic

  1. #1
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    Prove that a quotient group is cyclic

    If G is a cyclic group, prove that G/N is cyclic, where N is any subgroup of G.

    I first started out stating that since G is a cyclic group, there exists a generator x \epsilon G such that <x> = G. So I would need to show that <Nx> = G.

    I don't know where exactly to continue from this point to finish the proof. Any ideas on how?

    Thank you.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by crushingyen View Post
    If G is a cyclic group, prove that G/N is cyclic, where N is any subgroup of G.

    I first started out stating that since G is a cyclic group, there exists a generator x \epsilon G such that <x> = G. So I would need to show that <Nx> = G.

    I don't know where exactly to continue from this point to finish the proof. Any ideas on how?

    Thank you.
    Well, every elements of your group is of the form x^n for some n \in \mathbb{Z}, so every element of your quotient group looks like x^nN for some n \in \mathbb{Z}.

    What does this tell you?
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