Abstract Algebra – Infinite Abelian Group Has Element of Infinite Order

Question

Is the statement true?

Every infinite abelian group has at least one element of infinite order.

I am searching for an infinite abelian group with all elements having finite order.
Please help me to find such groups.

Answer 1

Let $G=\{z\in \mathbb{C}:z^n=1$ for some positive integer $n\}$. Then $G$ with complex multiplication is an infinite abelian group but every element has finite order.