Need some help on the following 2 related problems:

Show that any finitely generated subgroup of the additive group of rationals (Q,+,0) is cyclic.

Prove that this group is not isomorphic to the direct product of two copies of it.

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- Sep 6th 2009, 08:11 PMMathBirdcyclic group
Need some help on the following 2 related problems:

Show that any finitely generated subgroup of the additive group of rationals (Q,+,0) is cyclic.

Prove that this group is not isomorphic to the direct product of two copies of it. - Sep 6th 2009, 08:41 PMThePerfectHacker
Remember some facts about the structure of . For define to be the ideal generated these elements, i.e. . Since ideals are principal it means , where . We call the greatest common divisor of .

Now given, a finitely generated subgroup of . This means, by definition, . The subgroup generated by these elements is precisely where . Define, and . Thus, . Define to be the greatest common divisor of . Then, as seen in the first paragraph, . Thus, we see that must be cyclic.

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Prove that this group is not isomorphic to the direct product of two copies of it.

It is trivial to show that is not isomorphic to . - Sep 6th 2009, 09:21 PMMathBird
This is great!