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College Algebra Exam Review 96

College Algebra Exam Review 96 - until the line ` t...

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106 2. BASIC THEORY OF GROUPS Observe that the rotation r t through any angle t around the z –axis is a symmetry of the disk. Such rotations satisfy r t r s D r t C s and in particular r t r t D r 0 D e , where e is the nonmotion. It follows that N D f r t W t 2 R g is a subgroup of D . For any line passing through the origin in the .x; y/ –plane , the flip over that line (i.e., the rotation by about that line) is a symmetry of the disk, interchanging the top and bottom faces of the disk. Denote by j t the flip over the line ` t which passes through the origin and the point 2 4 cos .t/ sin .t/ 0 3 5 ; and write j D j 0 for the flip over the x –axis. Each j t generates a subgroup of D of order 2. Symmetries of the disk are illustrated in Figure 2.3.1 . j r Figure 2.3.1. Symmetries of the disk. Next, we observe that each j t can be expressed in terms of j and the rotation r t . To perform the flip j t about the line
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Unformatted text preview: until the line ` t overlays the x –axis, then perform the ﬂip j over the x – axis, and ﬁnally rotate the disk so that ` t is returned to its original position. Thus j t D r t jr ± t , or j t r t D r t j . Therefore, we need only work out how to compute products involving the ﬂip j and the rotations r t . Note that j applied to a point 2 4 ² cos .s/ ² sin .s/ 3 5 in the disk is 2 4 ² cos . ± s/ ² sin . ± s/ 3 5 , and r t applied to 2 4 ² cos .s/ ² sin .s/ 3 5 is 2 4 ² cos .s C t/ ² sin .s C t/ 3 5 : In the Exercises, you are asked to verify the following facts about the group D : 1. jr t D r ± t j , and j t D r 2t j D jr ± 2t . 2. All products in D can be computed using these relations....
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