Paper 3, Section II, D

State and prove the orbit-stabilizer theorem.

Let $G$ be the group of all symmetries of a regular octahedron, including both orientation-preserving and orientation-reversing symmetries. How many symmetries are there in the group $G$ ? Let $D$ be the set of straight lines that join a vertex of the octahedron to the opposite vertex. How many lines are there in the set $D$ ? Identify the stabilizer in $G$ of one of the lines in $D$.

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