Paper 3, Section II, D

Groups | Part IA, 2021

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

(b) Let nk1n \geqslant k \geqslant 1 be integers. Let G=SnG=S_{n}, the symmetric group of degree nn, and XX be the set of all ordered kk-tuples (x1,,xk)\left(x_{1}, \ldots, x_{k}\right) with xi{1,2,,n}x_{i} \in\{1,2, \ldots, n\}. Then GG acts on XX, where the action is defined by σ(x1,,xk)=(σ(x1),,σ(xk))\sigma\left(x_{1}, \ldots, x_{k}\right)=\left(\sigma\left(x_{1}\right), \ldots, \sigma\left(x_{k}\right)\right) for σSn\sigma \in S_{n} and (x1,,xk)X\left(x_{1}, \ldots, x_{k}\right) \in X. For x=(1,2,,k)Xx=(1,2, \ldots, k) \in X, determine Orb(x)\operatorname{Orb}(x) and Stab(x)\operatorname{Stab}(x) and verify that the orbit-stabiliser theorem holds in this case.

(c) We say that GG acts doubly transitively on XX if, whenever (x1,x2)\left(x_{1}, x_{2}\right) and (y1,y2)\left(y_{1}, y_{2}\right) are elements of X×XX \times X with x1x2x_{1} \neq x_{2} and y1y2y_{1} \neq y_{2}, there exists some gGg \in G such that gx1=y1g x_{1}=y_{1} and gx2=y2g x_{2}=y_{2}.

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

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