# \mathbb{Q}\times\mathbb{Q} is not cyclic

• Oct 8th 2011, 07:45 PM
dwsmith
\mathbb{Q}\times\mathbb{Q} is not cyclic
Prove that $\mathbb{Q}\times\mathbb{Q}$ is not cyclic.

How do I do this my book says nothing.
• Oct 8th 2011, 07:49 PM
Drexel28
Re: \mathbb{Q}\times\mathbb{Q} is not cyclic
Quote:

Originally Posted by dwsmith
Prove that $\mathbb{Q}\times\mathbb{Q}$ is not cyclic.

How do I do this my book says nothing.

If $\mathbb{Q}^2$ was cyclic, then $\mathbb{Q}$ being the image of $\mathbb{Q}^2$ under the projection mapping would be cyclic, so it suffices to show that $\mathbb{Q}$ is not cyclic. To see this suppose that there was an isomorphism $f:\mathbb{Q}\to\mathbb{Z}$ we see then $f(1)=f(m\frac{1}{m})=mf(\frac{1}{m})$ so that $f(1)$ is divisible by $m$ for every $m\in\mathbb{Z}$ and so $f(1)=0$, but since $f(0)=0$ this contradicts injectivity of $f$.
• Oct 10th 2011, 02:25 AM
Swlabr
Re: \mathbb{Q}\times\mathbb{Q} is not cyclic
Quote:

Originally Posted by Drexel28
If $\mathbb{Q}^2$ was cyclic, then $\mathbb{Q}$ being the image of $\mathbb{Q}^2$ under the projection mapping would be cyclic, so it suffices to show that $\mathbb{Q}$ is not cyclic.

Can;t you just say "subgroups of cyclic groups are cyclic"...
• Oct 10th 2011, 12:48 PM
roninpro
Re: \mathbb{Q}\times\mathbb{Q} is not cyclic
To show that $\mathbb{Q}$ is not cyclic, it seems conceptually more straightforward to take an arbitrary number $m/n$ (say, in lowest terms) and say that it can never meet $1/p$, where $p$ is some prime that is not a factor of $n$. (For example, no multiples of $3/16$ will ever be $1/5$.) So no single number can generate all of $\mathbb{Q}$.
• Oct 10th 2011, 02:22 PM
Drexel28
Re: \mathbb{Q}\times\mathbb{Q} is not cyclic
Quote:

Originally Posted by Swlabr
Can;t you just say "subgroups of cyclic groups are cyclic"...

Sure, either way works.

Quote:

Originally Posted by roninpro
To show that $\mathbb{Q}$ is not cyclic, it seems conceptually more straightforward to take an arbitrary number $m/n$ (say, in lowest terms) and say that it can never meet $1/p$, where $p$ is some prime that is not a factor of $n$. (For example, no multiples of $3/16$ will ever be $1/5$.) So no single number can generate all of $\mathbb{Q}$.

Another good way to look at it! My method shows more generally that any divisible group is not cyclic.