Is $\mathbb{Q}/\mathbb{Z}$ isomorphic to $\mathbb{Q}$?
My guess is no. Does the first isomorphism theorem have anything to do with this?
Any hints appreciated, thanks.
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Is $\mathbb{Q}/\mathbb{Z}$ isomorphic to $\mathbb{Q}$?
My guess is no. Does the first isomorphism theorem have anything to do with this?
Any hints appreciated, thanks.
HINT: What is the order of $\frac12+\Bbb Z$ in $\Bbb Q/\Bbb Z$? Does that match anything in $\Bbb Q$?
Hint: $\Bbb Q$ is an ordered group, in particular $0<\frac12<1$. But in $\Bbb{Q/Z}$ we have that $$\left(\frac12+\Bbb Z\right)+\left(\frac12+\Bbb Z\right)=0+\Bbb Z=1+\Bbb Z.$$
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HINT: What is the order of $\frac12+\Bbb Z$ in $\Bbb Q/\Bbb Z$? Does that match anything in $\Bbb Q$?
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