Every infinite abelian group has at least one element of infinite order? Is the statement true? > Every innite abelian group has at least one element of innite order. I am searching for an infinite abelian group with all elements having finite order. Please help me to find such groups.

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.