3
$\begingroup$

I need to find what are the at the group $\mathbb{Q}/\mathbb{Z}$.
I think that any element at this group has a finite order, but I don't know how to prove it... I'd like to get help with the proof writing...

If I'm wrong, I'd like to to know it too...

BTW:
$\mathbb{Z}=(\mathbb{Z},+)$
$\mathbb{Q}=(\mathbb{Q},+)$

Thank you!

$\endgroup$
  • 1
    $\begingroup$ This is a nice example that shows the existence of an infinite group in which every element has a finite order :) $\endgroup$ – Prism Dec 19 '13 at 7:44
6
$\begingroup$

Yes, you're correct that this is a group in which every element has finite order. Since you've asked about proof writing, I'll write out essentially a full proof.


Choose some $\alpha \in \Bbb{Q} / \Bbb{Z}$. By definition of the quotient group, there is a rational number $r$ for which

$$\alpha = r + \mathbb{Z}$$

Now by the definition of $\mathbb{Q}$, there are integers $a$ and $b \ne 0$ (we take $b > 0$) without any loss of generality) for which

$$r = \frac a b \implies \alpha = \frac a b + \mathbb{Z}$$

Therefore,

$$b\alpha = b\left(\frac a b + \Bbb Z\right) = b \frac a b + \Bbb{Z} = a + \Bbb{Z}$$

But since $a$ is an integer, $a + \Bbb{Z} = \Bbb{Z}$ is the identity in $\Bbb{Q} / \Bbb{Z}$, so $\alpha$ has finite order (in fact, the order is at most $b$) as desired.

$\endgroup$
4
$\begingroup$

If you know the nature of Prüfer group then you may use this fact that:

$$\mathbb Q/\mathbb Z=\sum_p\mathbb Z(p^{\infty})$$ However @T.Bongers's post makes the problem easy to understand step by step. Also, it is good to know that, the Prüfer group is an infinite $p-$primary group whose every subgroups if finite and indeed cyclic.

$\endgroup$
  • $\begingroup$ I start my morning visiting my friends! +1 $\endgroup$ – Namaste Dec 19 '13 at 11:57
  • $\begingroup$ @amwhy: it is so early my friend :-) $\endgroup$ – mrs Dec 19 '13 at 12:01
  • $\begingroup$ It is 6:00 am, my time. Not super early, but early! ;-) $\endgroup$ – Namaste Dec 19 '13 at 12:01
  • 1
    $\begingroup$ My current goal is to earn a gold star in abstract algebra and group theory, though elementary set theory is "up there" too. I did earn a gold star (1000 upvotes) in logic most recently! I see your "top" badge (>750 upvotes) is in "group theory". Good for you! We'll get you to gold, yet!! $\endgroup$ – Namaste Dec 19 '13 at 12:03
  • 1
    $\begingroup$ And I for you, dear friend! You deserve "Gold for Groups"!! ;-) $\endgroup$ – Namaste Dec 19 '13 at 12:15
2
$\begingroup$

Each element of $G=(\mathbb{Q}/\mathbb{Z},+)$ is of finite order.

$x\in G$, Then $x={p\over q}+\mathbb{Z}, p\in\mathbb{Z},q\in\mathbb{N}$

what is $qx$ then?

$qx=({p\over q}+\mathbb{Z})+({p\over q}+\mathbb{Z})+\dots$ $q$ times$=p+\mathbb{Z}=\mathbb{Z}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.