HW5_Solutions

# HW5_Solutions - Math 541 Solutions to HW#5 The following...

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Math 541 Solutions to HW #5 The following are from Gallian, Chapter 3 (6th edition). In this solution set, | a | denotes the order of the element a (i.e. ord( a )). # 1 : Z 12 : * | Z 12 | = 12; * | 0 | = 1; | 1 | = 12; | 2 | = 6; | 3 | = 4; | 4 | = 3; | 5 | = 12; | 6 | = 2; | 7 | = 12; | 8 | = 3; | 9 | = 4; | 10 | = 6; | 11 | = 12. U (10): * | U (10) | = φ (10) = φ (2 · 5) = (1)(4) = 4; * | 1 | = 1; | 3 | = 4; | 7 | = 4; | 9 | = 2. U (12): * | U (12) | = φ (12) = φ (2 2 · 3) = (2)(2) = 4; * | 1 | = 1; | 5 | = 2; | 7 | = 2; | 11 | = 2. U (20): * | U (20) | = φ (20) = φ (2 2 · 5) = (2)(4) = 8; * | 1 | = 1; | 3 | = 4; | 7 | = 4; | 9 | = 2; | 11 | = 2; | 13 | = 4; | 17 | = 4; | 19 | = 2. D 4 : * | D 4 | = 8; * | R 0 | = 1; | R 90 | = 4; | R 180 | = 2; | R 270 | = 4; | H | = 2; | V | = 2; | D | = 2; | D’ | = 2. Observations : The order of every element divides the order of the group. # 2 : In Q , we have h 1 2 i = { 0 , ± 1 2 , ± 1 , ± 3 2 , ± 2 , ± 5 2 ,... } In Q * , we have h 1 2 i = { ...

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HW5_Solutions - Math 541 Solutions to HW#5 The following...

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