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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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    If $x$ generates $G$, then does its image $\bar x$ in $G/H$ generate $G/H$?2012-01-23
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    I would think so?2012-01-23
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    @Jackson: If you actually think so, then try proving it!2012-01-23
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    Could I say something like x + h element of G, then G = , then x=g*n for some n2012-01-23
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    I then would like to say (g+H)*n = gn+Hn2012-01-23
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    Do you mean to write that $x + H$ is an element of $G/H$?2012-01-23
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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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    Why does the 2 get absorbed into the H?2012-01-23
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    @Jackson: I'm not sure what you mean. It is just the definition of coset multiplication: $g^2 + H = (g + g) + H = (g + H) \cdot (g + H) = (g + H)^2$. It is defined as $(a + H) \cdot (b + H) = (a + b) + H$. Does that help?2012-01-23
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    It might be a bit confusing that you use the notation $g^2$ when you have written G additively.2012-01-23
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    @m.k.: Aargh! Don't mix multiplicative and additive notation! If you write $a+H$ for cosets, you are implicitly saying that you will write $G$ additively; if you write $g^2$, you are implicitly saying that you will write it multiplicatively (unless you happen to be working in a ring, where both addition and multiplication make sense). If you are going to use $+$ for the operation in $G$ and $\cdot$ for the operation in $H$, then $(g+H)^n = ng+H$.2012-01-23
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    @Arturo (and Tobias): I think you are right, it's more clear that way. I'm used to always writing $g^n$ in groups for some reason..2012-01-23
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    @m.i.: To be perfectly honest, I'm non too happy with this, but at least it doesn't mix in the two operations inside of $G$. If I wanted to draw a clear distinction between the operations in $G$ and in $H$, I would normally use $+$ and $\oplus$, or $\cdot$ and $\odot$, but at least what you've written now doesn't make it seem like $G$ has two operations...2012-01-23
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    Ok, I have some comments/questions: For coset multiplication, are you saying (g+H)^n yields gn+H? I feel like (g+H)*n would yield gn+H. So g+H generates G/H which means its cyclic2012-01-23
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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 $\varphi(G)=\langle \varphi(g)\rangle$.