Paper 3, Section II, E

Groups | Part IA, 2012

(i) State and prove the Orbit-Stabilizer Theorem.

Show that if GG is a finite group of order nn, then GG is isomorphic to a subgroup of the symmetric group SnS_{n}.

(ii) Let GG be a group acting on a set XX with a single orbit, and let HH be the stabilizer of some element of XX. Show that the homomorphism GSym(X)G \rightarrow \operatorname{Sym}(X) given by the action is injective if and only if the intersection of all the conjugates of HH equals {e}\{e\}.

(iii) Let Q8Q_{8} denote the quaternion group of order 8 . Show that for every n<8,Q8n<8, Q_{8} is not isomorphic to a subgroup of SnS_{n}.

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