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Which of the following statements are true:
1. every countable group G has only countably many distinct subgroups.
2. any automorphism of the group $\mathbb{Q}$ under addition is of the form $x→qx$ for some $q\in\mathbb{Q}$
3. all non-trivial proper subgroups of $(\mathbb{R},+)$ are cyclic.
4. every infinite abelian group has at least one element of infinite order.
5. there is an element of order $51$ in the multiplicative group $(\mathbb{Z}/103\mathbb{Z})^*$

My thoughts:
1. true as union of uncountable number of countable set is uncountable
2. true as any homomorphism must be one of those form
3. false as $(\mathbb{Q},+)$ is not cyclic.
4. false example circle group.
5. true by Fermat's little theorem.

Are my guesses correct?

  • 2
    5 is true, but I don't think Fermat is involved. 4 is false, but I don't know what the circle group is --- if its the group of rotations of the circle, it certainly has elements of infinite order.2012-12-10
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    Your justification of (1) is weird, to say the least: what has that to do with the question asked?. (2) This is no mathematical justification at all. (3) Good. (4) Not good, as the circle group has lots of elements of infinite order. You can take the Prufer group, though. (5) Well, yes...but you can't apply Cauchy's Theorem *directly* as $\,51\,$ is not a prime2012-12-10

2 Answers 2