Paper 3, Section II, D

Groups | Part IA, 2013

(a) Let GG be a finite group. Show that there exists an injective homomorphism GSym(X)G \rightarrow \operatorname{Sym}(X) to a symmetric group, for some set XX.

(b) Let HH be the full group of symmetries of the cube, and XX the set of edges of the cube.

Show that HH acts transitively on XX, and determine the stabiliser of an element of XX. Hence determine the order of HH.

Show that the action of HH on XX defines an injective homomorphism HSym(X)H \rightarrow \operatorname{Sym}(X) to the group of permutations of XX, and determine the number of cosets of HH in Sym(X)\operatorname{Sym}(X).

Is HH a normal subgroup of Sym(X)?\operatorname{Sym}(X) ? Prove your answer.

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