College Algebra Exam Review 97

# College Algebra Exam Review 97 - Figure 2.3.2 for the case...

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2.3. THE DIHEDRAL GROUPS 107 3. The symmetry group D of the disk consists of the rotations r t for t 2 R and the ﬂips j t D r 2t j . Writing N D f r t W t 2 R g , we have D D N [ Nj . 4. The subgroup N of D satisﬁes aNa ± 1 D N for all a 2 D . Next, we turn to the symmetries of the regular polygons. Consider a regular n -gon with vertices at 2 4 cos .2k±=n/ sin .2k±=n/ 0 3 5 for k D 0;1;:::; n ± 1 . Denote the symmetry group of the n -gon by D n . In the exercises, you are asked to verify the following facts about the symmetries of the n -gon: 1. The rotation r D r 2±=n through an angle of 2±=n about the z axis generates a cyclic subgroup of D n of order n . 2. The “ﬂips” j k±=n D r k2±=n j D r k j , for k 2 Z , are symme- tries of the n -gon. 3. The distinct ﬂip symmetries of the n -gon are r k j for k D 0;1;:::;n ± 1 . 4. If n is odd, then the axis of each of the ﬂips passes through a vertex of the n -gon and the midpoint of the opposite edge. See
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Unformatted text preview: Figure 2.3.2 for the case n D 5 . 5. If n is even and k is even, then j k±=n D r k j is a ﬂip about an axis passing through a pair of opposite vertices of the n-gon. 6. If n is even and k is odd, then j k±=n D r k j is a ﬂip about an axis passing through the midpoints of a pair of opposite edges of the n-gon. See Figure 2.3.2 for the case n D 6 . rj j r j rj r Figure 2.3.2. Symmetries of the pentagon and hexagon. The symmetry group D n consists of the 2n symmetries r k and r k j , for ² k ² n ± 1 . It follows from our computations for the symmetries of the disk that jr D r ± 1 j , so jr k D r ± k j for all k . This relation allows the computation of all products in D n ....
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