Rotating a Square The square ABCD below has a pin through

Chapter 10, Problem 818

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Rotating a Square The square ABCD below has a pin through it at point P so that the square can be rotated in a clockwise direction. The mathematical system consists of the set 5A, B, C, D6 and the operation 3. The elements of the set represent different rotations of the square clockwise, as follows. A = rotate about the point P clockwise 90 B = rotate about the point P clockwise 180 C = rotate about the point P clockwise 270 D = rotate about the point P clockwise 360 The operation 3 means followed by. For example, A 3 B means a clockwise rotation of 90 followed by a clockwise rotation of 180 or a clockwise rotation of 270. Since C represents a clockwise rotation of 270, we write A 3 B = C. This result and two other results are shown in the following table. A B C P D Complete the table and determine if the mathematical system is a commutative group. Explain