Mathematic Methods HW Solutions 17

Mathematic Methods HW Solutions 17 - Chapter 3 17 13.5 The...

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Unformatted text preview: Chapter 3 17 13.5 The 4s group 13.6 The cyclic group 13.7 The 4s group 13.10 If R = 90 rotation, P = reflection through the y axis, and Q = PR, then the 8 matrices of the symmetry group of the square are: I = parenleftbigg 1 1 parenrightbigg , R = parenleftbigg- 1 1 parenrightbigg , R 2 = parenleftbigg- 1- 1 parenrightbigg =- I, R 3 = parenleftbigg 1- 1 0 parenrightbigg =- R, P = parenleftbigg- 1 0 1 parenrightbigg , PR = parenleftbigg 0 1 1 0 parenrightbigg = Q, PR 2 = parenleftbigg 1- 1 parenrightbigg =- P, PR 3 = parenleftbigg- 1- 1 parenrightbigg =- Q, with multiplication table: I R- I- R P Q- P- Q I I R- I- R P Q- P- Q R R- I- R I- Q P Q- P- I- I- R I R- P- Q P Q- R- R I R- I Q- P- Q P P P Q- P- Q I R- I- R Q Q- P- Q P- R I R- I- P- P- Q P Q- I- R I R- Q- Q P Q- P R- I- R I 13.11 The 4 matrices of the symmetry group of the rectangle are I = parenleftbigg 1 1 parenrightbigg , P = parenleftbigg- 1 0 0 1 parenrightbigg , parenleftbigg 1- 1 parenrightbigg =- P, parenleftbigg...
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