Exercise 15, § 2

Let $X$ be a compact metrizable space, let $\Gamma$ be the group of all homeomorphisms of $X$, and let $G$ be a subgroup of $\Gamma$ which operates transitively on $X$. If $H$ is the centralizer of $G$ in $\Gamma$, show that $H$ is equicontinuous. [Argue by reductio ad absurdum, supposing that $H$ is not equicontinuous at some point $a \in X$; deduce the existence of a sequence of points $x_n \in X$ and a sequence of elements $u_n \in H$ such that

where $b \neq c$. Hence show that the sequence $(u_n)$ converges pointwise in $X$, but that no point of $X$ is a point of uniform convergence of this sequence; this contradicts Exercise 9 of § 1.]