Explain what is meant by a covering projection. State and prove the pathlifting property for covering projections, and indicate briefly how it generalizes to a lifting property for homotopies between paths. [You may assume the Lebesgue Covering Theorem.]

Let $X$ be a simply connected space, and let $G$ be a subgroup of the group of all homeomorphisms $X \rightarrow X$. Suppose that, for each $x \in X$, there exists an open neighbourhood $U$ of $x$ such that $U \cap g[U]=\emptyset$ for each $g \in G$ other than the identity. Show that the projection $p: X \rightarrow X / G$ is a covering projection, and deduce that $\Pi_{1}(X / G) \cong G$.

By regarding $S^{3}$ as the set of all quaternions of modulus 1 , or otherwise, show that there is a quotient space of $S^{3}$ whose fundamental group is a non-abelian group of order $8 .$