Question on Groups

Question

Give an example of an infinite group in which every element has a finite order

Answer 1

$\Bbb{Z_2 \oplus \Bbb{Z_2 \oplus \Bbb{Z_2 \oplus \dots}}}$

To make it more interesting, explore on this problem-

If G is a finitely generated, torsion group, then is G necessarily finite?

Answer 2

Take the sum of countably many copies of $\mathbb{Z}/2\mathbb{Z}$, that is, the group of sequences of elements of $\mathbb{Z}/2\mathbb{Z}$ that have only finitely many non-zeros. Each element in this group has order 2.

As another example, take the group $\{z \in \mathbb{C} \mid z = e^{2\pi i x} , x \in \mathbb{Q} \}$, which is isomorphic to $\mathbb{Q}/\mathbb{Z}$. If $z = e^{2\pi i \frac{p}{q}}$, with $p$ and $q$ coprime, then $z$ has order $q$.