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Possible Duplicate:
$G$ modulo $N$ is a cyclic group when $G$ is cyclic

Prove that if $H$ is a subgroup of a cyclic group $G$, then $G/H$ must also be cyclic.

I think that I start off saying something like "$x+h$ is an element of $G$", but am not sure if this is a good start.

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    @m.k.: It is a duplicate, but the older question doesn’t really have an answer, so I’m unwilling to close this one.2012-01-23

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Suppose $(G, +)$ is a cyclic group. You can see that in the group $(G/H, \cdot)$ we have

$2g + H = (g + H) \cdot (g + H) = (g + H)^2$

$3g + H = (g + H) \cdot (g + H) \cdot (g + H) = (g + H)^3$

...

$ng + H = (g + H)^n$

Then use this to prove that $G/H$ is cyclic.

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    I see, since we have ng+H = (g+H)^n, we an say that g+H generates G/H and so G/H has a generator which means its cyclic.2012-01-23
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More precisely, if $G$ is cyclic and $\varphi:G\to\Gamma$ is a homomorphism then $\varphi(G)$ is a cyclic. Indeed, if $G=\langle g \rangle$ then $\varphi(G)=\langle \varphi(g)\rangle$.