Show that the group $(\Bbb R,+)/(\Bbb Z,+)$ has non-trivial finite subgroups.
The group is
$$(\Bbb R,+)/(\Bbb Z,+) = \{a + \Bbb Z \mid a \in \Bbb R\},$$
but I think all of the cosets are infinite? For example with $3+\Bbb Z = \{\dots,3,4,5,6, \dots \}$.
How can I get a finite subgroup when the cosets are of the form $a + \Bbb Z$ for $a \in \Bbb R$?
Best Answer
Hint: Let $n\in\Bbb N.$ We have
$$n\left(\frac{1}{n}+\Bbb Z\right)=\Bbb Z.$$