Torsion group

Remarks

Example: the group $\Q/\Z$, the rational numbers modulo the integers under addition. If $x = \tfrac{a}{b} + \Z$ with $a, b \in \Z$ and $b \ge 1$, then $b x = a + \Z = 0$, so $x$ has finite order. The Prüfer group is another infinite example.

Non-example: the group $\Z$ under addition. For $n \ne 0$ and $k \ge 1$, $k n \ne 0$, so $n$ has infinite order.

A group of finite exponent is a torsion group, but $\Q/\Z$ is a torsion group of infinite exponent. A finitely generated abelian torsion group is finite. For nonabelian groups this fails: there are finitely generated infinite torsion groups, such as the Grigorchuk group; see the Burnside problem.