Paper 3, Section II, D

(a) Let $G$ be a finite group acting on a set $X$. For $x \in X$, define the orbit $\operatorname{Orb}(x)$ and the stabiliser $\operatorname{Stab}(x)$ of $x$. Show that $\operatorname{Stab}(x)$ is a subgroup of $G$. State and prove the orbit-stabiliser theorem.

(b) Let $n \geqslant k \geqslant 1$ be integers. Let $G=S_{n}$, the symmetric group of degree $n$, and $X$ be the set of all ordered $k$-tuples $\left(x_{1}, \ldots, x_{k}\right)$ with $x_{i} \in\{1,2, \ldots, n\}$. Then $G$ acts on $X$, where the action is defined by $\sigma\left(x_{1}, \ldots, x_{k}\right)=\left(\sigma\left(x_{1}\right), \ldots, \sigma\left(x_{k}\right)\right)$ for $\sigma \in S_{n}$ and $\left(x_{1}, \ldots, x_{k}\right) \in X$. For $x=(1,2, \ldots, k) \in X$, determine $\operatorname{Orb}(x)$ and $\operatorname{Stab}(x)$ and verify that the orbit-stabiliser theorem holds in this case.

(c) We say that $G$ acts doubly transitively on $X$ if, whenever $\left(x_{1}, x_{2}\right)$ and $\left(y_{1}, y_{2}\right)$ are elements of $X \times X$ with $x_{1} \neq x_{2}$ and $y_{1} \neq y_{2}$, there exists some $g \in G$ such that $g x_{1}=y_{1}$ and $g x_{2}=y_{2}$.

Assume that $G$ is a finite group that acts doubly transitively on $X$, and let $x \in X$. Show that if $H$ is a subgroup of $G$ that properly contains $\operatorname{Stab}(x)($ that is, $\operatorname{Stab}(x) \subseteq H$ but $\operatorname{Stab}(x) \neq H)$ then the action of $H$ on $X$ is transitive. Deduce that $H=G$.