262 5. ACTIONS OF GROUPS 5.6.5. Let G be the rotation group of the cube, acting on the set of faces of the cube. Show that map F 7! Stab .F/ is 2-to-1, from the set of faces to the set of stabilizer subgroups of faces. 5.6.6. Let G D S n and H D Stab .n/ Š S n ± 1 . Show that H is its own normalizer, so that the cosets of H correspond 1-to-1 with conjugates of H . Describe the conjugates of H explicitly. 5.6.7. Identify the group G of rotations of the cube with S 4 , via the action on the diagonals of the cube. G also acts transitively on the set of set of three 4–fold rotation axes of the cube; this gives a homomorphism of S 4 into S 3 . (a) Compute the resulting homomorphism of S 4 to S 3 explicitly. (For example, compute the image of a set of generators of S 4 .) Show that is surjective. Find the kernel of . (b) Show that the stabilizer of each 4–fold rotation axis is conjugate to D 4 ± S 4 . (c)
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