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Are the additive groups $(\mathbb Z, +)$ and $(\mathbb Q, +)$ isomorphic?

I know the group of integers is cyclic because $1$ generates all elements in $\mathbb Z$ , so $\langle1\rangle = \mathbb Z$.

However, I am having a hard time understanding why $\mathbb Q$ is not cyclic. I know they are not isomorphic, but can anyone help me understand why $\mathbb Q$ is not cyclic. Thank you.

Brian Fitzpatrick
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user1456
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  • Suppose $\mathbb{Q}$ were cyclic with some positive generator $g$. Think about a smaller rational number. –  Feb 20 '14 at 02:13
  • @thomas whoops. misread the problem! – Brian Fitzpatrick Feb 20 '14 at 02:21
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    A different hint: for every $q\in\mathbb{Q}$ and $m\in\mathbb{Z}$, there exists $r\in\mathbb{Q}$ such that $mr=q$. Show that this property would be preserved by a group homomorphism, and that it doesn't occur in $\mathbb{Z}$. – rfauffar Feb 20 '14 at 02:21

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If $(\mathbb{Q}, +)$ was cyclic, then you would have an element $\frac{a}{b}$ such that

$$\mathbb{Q} = \langle \frac{a}{b} \rangle.$$

That is, every rational number would be a positive integer multiple of $\frac{a}{b}$.

So you would, for example, have that there is an $n\in \mathbb{N}$ such that

$$n\frac{a}{b} = \frac{1}{b}.$$ That means that $na =1$ which means $a=1$.

So now note the existence of integer $m$ such that

$$m\frac{1}{b} = \frac{1}{2b}.$$ But the only way this can be true is if $m = \frac{1}{2}$.

So $(\mathbb{Q}, +)$ is not cyclic.

Thomas
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