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Thread: Prove that a quotient group is cyclic

  1. #1
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    Prove that a quotient group is cyclic

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

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

    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 $\displaystyle G$ is a cyclic group, prove that $\displaystyle G/N$ is cyclic, where $\displaystyle N$ is any subgroup of $\displaystyle G$.

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

    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 $\displaystyle x^n$ for some $\displaystyle n \in \mathbb{Z}$, so every element of your quotient group looks like $\displaystyle x^nN$ for some $\displaystyle n \in \mathbb{Z}$.

    What does this tell you?
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