Paper 3, Section II, D

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

(b) Let $H$ be the full group of symmetries of the cube, and $X$ the set of edges of the cube.

Show that $H$ acts transitively on $X$, and determine the stabiliser of an element of $X$. Hence determine the order of $H$.

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

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

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