Remark 3

Suppose that G is a discrete topological group. The subgroup $\iota (G)$ is then a discrete subgroup of $G^*$. In fact, the topology of G is defined by the distance $d$ on G given by $d(g, g') = 1$ if $g=\not g'$ and $d(g, g') = 0$ otherwise. The assertion then follows from the preceding remark.