Paper 3, Section II, E

(i) State and prove the Orbit-Stabilizer Theorem.

Show that if $G$ is a finite group of order $n$, then $G$ is isomorphic to a subgroup of the symmetric group $S_{n}$.

(ii) Let $G$ be a group acting on a set $X$ with a single orbit, and let $H$ be the stabilizer of some element of $X$. Show that the homomorphism $G \rightarrow \operatorname{Sym}(X)$ given by the action is injective if and only if the intersection of all the conjugates of $H$ equals $\{e\}$.

(iii) Let $Q_{8}$ denote the quaternion group of order 8 . Show that for every $n<8, Q_{8}$ is not isomorphic to a subgroup of $S_{n}$.

*Typos? Please submit corrections to this page on GitHub.*