Are all torsion groups finite groups?

Question

Are all torsion groups finite groups?

I've been trying to find a counter example, but have had no luck so far. Can anyone throw me one, or give me an idea to prove this?

Answer 1

Take infinite direct sum of finite abelian groups. For finitely generated examples google "Burnside problem".

If you do not know what is the direct sum of groups, take the group of roots of unity: $$ \{ e^{i\pi r}: r\in {\mathbb Q}\}. $$

Answer 2

Hmm, how about $\mathbb Q/\mathbb Z$?