Math 761 Fall 2015
نویسنده
چکیده
(a) Describe the conjugacy classes of D and use this to prove that D is simple. [Hint: Any normal subgroup is a union of conjugacy classes.] (b) Compute the number of distinguishable ways to color the faces of a dodecahedron with k colors. [Hint: Let X be the set of all colorings, so that |X| = k12. Many of these colorings are indistinguishable after rotation so we really want to know the number of orbits |X/D|. Use part (a) and Burnside’s Lemma.] (c) Prove that D is isomorphic to the alternating group A5. [Hint: There are five cubes that can be inscribed in a dodecahedron. The action of D defines a nontrivial homomorphism φ : D → S5. Composing this with the “sign” homomorphism σ : S5 → {±1} gives a homomorphism σφ : D → {±1}. Since D is simple the first homomorphism must be injective and the second must be trivial.]
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