# Basic group theory question on countable groups and infinite abelian groups.

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.

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Hint: these aren't very obscure possibilities. You can do both with vector spaces. –  Chris Eagle May 6 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) $\mathbb{Q}/\mathbb{Z}$.