Paper 3, Section II, D

Groups | Part IA, 2015

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

(a) Prove that the symmetric group SnS_{n} acts on the set P(n)P^{(n)} of all polynomials in nn variables x1,,xnx_{1}, \ldots, x_{n}, if we define σf\sigma \cdot f to be the polynomial given by

(σf)(x1,,xn)=f(xσ(1),,xσ(n))(\sigma \cdot f)\left(x_{1}, \ldots, x_{n}\right)=f\left(x_{\sigma(1)}, \ldots, x_{\sigma(n)}\right)

for fP(n)f \in P^{(n)} and σSn\sigma \in S_{n}. Find the orbit of f=x1x2+x3x4P(4)f=x_{1} x_{2}+x_{3} x_{4} \in P^{(4)} under S4S_{4}. Find also the order of the stabiliser of ff.

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

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