College Algebra Exam Review 213

College Algebra Exam Review 213 - hedron are isomorphic to...

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4.2. ROTATIONS OF THE DODECAHEDRON AND ICOSAHEDRON 223 Figure 4.2.5. Cube inscribed in the dodecahedron. Now, it remains to show that the image of the rotation group in S 5 is the group of even permutations A 5 . We could do this by explicit com- putation. However, by using a previous result, we can avoid doing any computation at all. We established earlier that for each n , A n is the unique subgroup of S n of index 2 (Exercise 2.5.16 ). Since the image of the rota- tion group has 60 elements, it follows that it must be A 5 . Proposition 4.2.1.
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Unformatted text preview: hedron are isomorphic to the group of even permutations A 5 . Exercises 4.2 4.2.1. Show that no rotation of the dodecahedron leaves each of the five inscribed cubes fixed. Thus the action of the rotation group on the set of inscribed cubes induces an injective homomorphism of the rotation group into S 5 . 4.2.2. Let A D f 2 4 cos 2k±=5 sin 2k±=5 1=2 3 5 W 1 ± k ± 5 g and B D f 2 4 cos .2k C 1/±=5 sin .2k C 1/±=5 ² 1=2 3 5 W 1 ± k ± 5 g :...
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