An often given example of a group of infinite order where every element has infinite order is the group $\dfrac{\mathbb{(Q, +)}}{(\mathbb{Z, +})}$.
But I don't see why every element necessarily has finite order in this group. Why is this true?
Also, what is the identity element of this group?
Best Answer
The identity of $G =\mathbb Q / \mathbb Z$ is $\mathbb Z$.
1) Every element of $G$ has finite order. Indeed, if you take any $r + \mathbb Z \in G$ where $r \in \mathbb Q$ then we may write $r = \frac{m}{n}$, with $m,n \in \mathbb Z $ and $n > 0$. Thus
$$n (r + \mathbb Z) = m + \mathbb Z = \mathbb Z$$
then it follows that $\left|r + \mathbb Z\right| \leq n$.
Edit:
2) The group has infinite order because we may choose an element in $G$ with arbitrarily large order, consider $\frac{1}{n} + \mathbb Z$, where $\left|\frac{1}{n} + \mathbb Z\right| = n$ (why?).