infinite order of element with element in an infinite group

Question

If $G$ is a infinite group, then $G$ must have an element of infinite order.

Is this true?

I know that if $G$ is infinite cyclic, then it's isomorphic to $\mathbb Z$.

(I guess fact is irrelevant now)

Answer 1

Consider $(\mathbb Z/2\mathbb Z)^{\mathbb N}$ or $\mathbb Q/\mathbb Z$.

Answer 2

in an abelian group the elements of finite order form its torsion subgroup. a torsion group is infinite iff it is not finitely generated. $\Bbb Q / \Bbb Z$, cited by Hagen in his answer is a particularly clear example.