Exercise 10, § 2

Let $G$ be a quasi-topological group in which every subset consisting of a single point is closed and whose identity component $C$ is such that $G/C$ is finite. Show that if $G/C$ has $k$ elements, then the set of conjugates $xax^{-1}$ of an element $a \in G$ is either infinite or else has at most $k$ elements (consider the mapping $x \to xax^{-1}$ of $G$ onto this set). In particular, if $G$ is connected, the set of conjugates of every element not in the centre of $G$ is infinite.