Paper 3, Section II, D

What does it mean for a group $G$ to act on a set $X$ ? For $x \in X$, what is meant by the orbit $\operatorname{Orb}(x)$ to which $x$ belongs, and by the stabiliser $G_{x}$ of $x$ ? Show that $G_{x}$ is a subgroup of $G$. Prove that, if $G$ is finite, then $|G|=\left|G_{x}\right| \cdot|\operatorname{Orb}(x)|$.

(a) Prove that the symmetric group $S_{n}$ acts on the set $P^{(n)}$ of all polynomials in $n$ variables $x_{1}, \ldots, x_{n}$, if we define $\sigma \cdot f$ to be the polynomial given by

$(\sigma \cdot f)\left(x_{1}, \ldots, x_{n}\right)=f\left(x_{\sigma(1)}, \ldots, x_{\sigma(n)}\right)$

for $f \in P^{(n)}$ and $\sigma \in S_{n}$. Find the orbit of $f=x_{1} x_{2}+x_{3} x_{4} \in P^{(4)}$ under $S_{4}$. Find also the order of the stabiliser of $f$.

(b) Let $r, n$ be fixed positive integers such that $r \leqslant n$. Let $B_{r}$ be the set of all subsets of size $r$ of the set $\{1,2, \ldots, n\}$. Show that $S_{n}$ acts on $B_{r}$ by defining $\sigma \cdot U$ to be the set $\{\sigma(u): u \in U\}$, for any $U \in B_{r}$ and $\sigma \in S_{n}$. Prove that $S_{n}$ is transitive in its action on $B_{r}$. Find also the size of the stabiliser of $U \in B_{r}$.

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