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sep08 - Math 3124 Thursday September 8 September 8 Ungraded...

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Unformatted text preview: Math 3124 Thursday, September 8 September 8 Ungraded Homework Problem 8.5 on page 50 Determine the group of symmetries of a regular pentagon. We will label the vertices of the regular pentagon 1, 2, 3, 4, 5 going clockwise. Denote by A, B, C, D, E the lines from 1, 2, 3, 4, 5 to the midpoint of the opposite edge, respectively. We note that we have at most 10 symmetries. Indeed there are at most 5 choices for A. Then B has to remain adjacent to A, so there are at most 2 choices for B. Once A and B are fixed, the whole pentagon is fixed. Therefore there are at most 5 * 2 = 10 symmetries. Thus once we have exhibited 10 symmetries, we will know that the order of the group of symmetries is exactly 10. The following table describes 10 symmetries. Symmetry Permutation Identity (1) 72 clockwise (1 2 3 4 5) 144 clockwise (1 3 5 2 4) 216 clockwise (1 4 2 5 3) 288 clockwise (1 5 4 3 2) reflect in A (2 5)(3 4) reflect in B (1 3)(4 5) reflect in C (2 4)(1 5) reflect in D (3 5)(1 2) reflect in E (1 4)(2 3) The Cayley table is then ◦ (1) (1 2 3 4 5) (1 3 5 2 4) (1 4 2 5 3) (1 5 4 3 2) (2 5)(3 4) (1 3)(4 5) (2 4)(1 5) (3 5)(1 2) (1 4)(2 3) (1) (1) (1 2 3 4 5) (1 3 5 2 4) (1 4 2 5 3) (1 5 4 3 2) (2 5)(3 4)...
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sep08 - Math 3124 Thursday September 8 September 8 Ungraded...

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