Finitely generated subgroup of Q

**roporte** · November 9th 2008, 10:17 AM

Let H be a subgroup of $\mathbb{Q}$ finitely generated. Prove that H is cyclic.

**NonCommAlg** · November 9th 2008, 03:34 PM

Originally Posted by roporte

Let H be a subgroup of $\mathbb{Q}$ finitely generated. Prove that H is cyclic.

you've been asking so many questions without showing us that you've thought about them for a minute! what makes me a little bit worried is that some of your questions are absolutely trivial

or just computational, which i've ignored, and some are fairly interesting! so, i really don't know if you basically have any idea about the questions?! i don't know about other members, who are

mostly nicer than me, but i'll keep helping you with those of your questions that are more interesting, however with less and less details! ok, regarding this question:

you should have mentioned that $\mathbb{Q}$ here is considered as an additive group, i.e. we want to prove that every finitely generated $\mathbb{Z}-$ submodule of $\mathbb{Q}$ is cyclic. suppose $H$ is a finitely generated

$\mathbb{Z}-$ submodule of $\mathbb{Q}.$ so: $H=\sum_{i=1}^n r_i \mathbb{Z},$ for some $r_i \in \mathbb{Q}.$ let $r_i=\frac{a_i}{b_i}, \ a_i, b_i \in \mathbb{Z},$ and: $\prod_{i=1}^n b_i= c.$ then: $H=\frac{1}{c} \sum_{i=1}^n \frac{a_ic}{b_i}\mathbb{Z}.$ now we have: $\sum_{i=1}^n \frac{a_ic}{b_i}\mathbb{Z}=b \mathbb{Z},$ for some $b \in \mathbb{Z}.$ why? thus: $H=\frac{b}{c}\mathbb{Z}. \ \ \Box$

**roporte** · November 9th 2008, 03:53 PM

First of all, Thanks for the answer!

I'm assisting to an algebra seminary in the math university here, buy i'm a magister in system engineer, simply i'm taking this seminary as a complement of my area of investigation related with generic programming that uses some algebra.

In the seminary we receive a lot of sheets of excercises and in this excercises i find some of them that i solve easily. But i also find excercises that i cannot solve and others that i can solve but i'm not sure about my solution. For that reason i ask here in the forum some of the excercises and you have helped me a lot really. i`m apologize for not explaining this situation before (i think it was unappropiate) or for any inconvinience, but belive me that i never ask a question if i hadn't thinked it before, and a lot.

Thanks again!

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