Exercise 9, § 8

Let $\Gamma$ be a finite group of homeomorphisms of a Hausdorff (resp. regular) space $X$, and let $R$ be the equivalence relation "there exists $\sigma \in \Gamma$ such that $y = \sigma(x)$" between $x$ and $y$ in $X$. Show that the quotient space $X/R$ is Hausdorff (resp. regular).