I am in need of examples of infinite groups such that all their respective elements are of finite order.

**Answer**

Here is one. Let $(\mathbb{Q},+)$ denote the groups of rational numbers under addition, and consider it’s subgroup $(\mathbb{Z},+)$ of integers. Then any element from the group $\mathbb{Q}/\mathbb{Z}$ has elements of the form $\frac{p}{q} + \mathbb{Z}$ which is of order at-most $q$. Hence it’s of finite order.

- Group of all roots of unity in $\mathbb{C}^{\times}.$

Here is a link from MathOverflow which might prove helpful.

**Attribution***Source : Link , Question Author : Joel , Answer Author : Community*