What does it mean for a group to act on a set ? For , what is meant by the orbit to which belongs, and by the stabiliser of ? Show that is a subgroup of . Prove that, if is finite, then .
(a) Prove that the symmetric group acts on the set of all polynomials in variables , if we define to be the polynomial given by
for and . Find the orbit of under . Find also the order of the stabiliser of .
(b) Let be fixed positive integers such that . Let be the set of all subsets of size of the set . Show that acts on by defining to be the set , for any and . Prove that is transitive in its action on . Find also the size of the stabiliser of .