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Prove that $\mathbb{Q}/\mathbb{Z}$ is divisible.

By definition, in a divisible group every element is the $k$th multiple of some other element. $\mathbb{Q}$ is a divisible basically by definition. It is an abelian group. And $\mathbb{Z}$ is a proper subgroup. I'm stuck.

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    To get you in the proper mind-set of how to prove this, try to prove more generally that any image of a divisible group is divisible.2012-01-30

1 Answers 1

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Let $G$ be a group, and let $N$ be a normal subgroup of $G$.

If $G$ is divisible, then for every $x\in G$ and for every $n\gt 0$ there exists a $y\in G$ such that $y^n = x$ (writing $G$ multiplicatively; if $G$ is written additively, then the equation would be $ny=x$).

Now, let $gN$ be an arbitrary element of $G/N$, and let $n\gt 0$. Can you prove that there exists $yN\in G/N$ such that $(yN)^n = gN$? (With $G$ written additively, given $g+N$ and $n\gt 0$, can you prove that there exists $y+N\in G/N$ such that $n(y+N) = g+N$?)

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    I see. Yes it is easy, as you said! thank you!2012-01-30