Here are two (maybe simple) questions.

A: Every countable group $G$ has only countably many distinct subgroups.

B: Every infinite abelian group has atleast one element of infinite order.

Both these statements are false. I am unable to find any counterexamples. Just Hints would be highly appreciated.

Thanks in advance.

  • 2
    $\begingroup$ Hint: these aren't very obscure possibilities. You can do both with vector spaces. $\endgroup$ – Chris Eagle May 6 '13 at 9:28


A) Consider the vector space $V$ over the field of two elements with a countably infinite basis. A countably infinite set has uncountably many subsets.

B) $\mathbb{Q}/\mathbb{Z}$.

  • 3
    $\begingroup$ You can recycle the example for A in B. $\endgroup$ – Hagen von Eitzen May 6 '13 at 9:30
  • 1
    $\begingroup$ Yes. I am all for recycling. But I also like some variety. $\endgroup$ – Jyrki Lahtonen May 6 '13 at 9:31
  • $\begingroup$ @ Jyrki Lahtonen: Thanks a lot. $\endgroup$ – user67773 May 6 '13 at 9:33

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