Solution: We will give an example of an infinite group where every element is of finite order. Consider the group of rational numbers \( \mathbb{Q} \) under addition and its subgroup \( \mathbb{Z} \). We claim that the quotient group \( \mathbb{Q} / \mathbb{Z} \) is an example of an infinite group where every element has finite order. Note that $\mathbb{Q} /\mathbb{Z} $ is a group under addition modulo \( \mathbb{Z} \) because $\mathbb{Z} $ is a normal subgroup of \( \mathbb{Q} \).

Note that \[ \mathbb{Q} /\mathbb{Z} = \left\{ \frac{m}{n} + \mathbb{Z} \mid m \in \mathbb{Z}, n \in \mathbb{N} \right\}. \] Since the representatives of \( \mathbb{Q} / \mathbb{Z} \) correspond to rational numbers within the interval \( [0,1) \), and there are infinitely many such rational numbers, it follows that \( \mathbb{Q} / \mathbb{Z} \) is an infinite group.

Now we will prove that every element of the group $\mathbb{Q} / \mathbb{Z} $ is of finite order. Let \( \frac{m}{n} + \mathbb{Z} \in \mathbb{Q} / \mathbb{Z} \). Multiplying by \( n \) gives: \[ n \cdot \left(\frac{m}{n} + \mathbb{Z} \right) = m + \mathbb{Z} = 0 + \mathbb{Z}. \] This shows that the order of \( \frac{m}{n} + \mathbb{Z} \) is at most \( n \). Since every element of \( \mathbb{Q} / \mathbb{Z} \) has finite order but the group itself is infinite, we conclude that \( \mathbb{Q} / \mathbb{Z} \) is an infinite group where every element has finite order.