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College Algebra Exam Review 99

# College Algebra Exam Review 99 - n-gon are r k j for k D...

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2.3. THE DIHEDRAL GROUPS 109 Exercises 2.3 2.3.1. Show that the elements j and r t of the symmetry group D of the disk satisfy the relations jr t D r t j , and j t D r 2t j D jr 2t . 2.3.2. 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 . (a) Writing N D f r t W t 2 R g , show that D D N [ Nj . (b) Show that all products in D can be computed using the relation jr t D r t j . (c) Show that the subgroup N of D satisfies aNa 1 D N for all a 2 D . 2.3.3. The symmetries of the disk are implemented by linear transforma- tions of R 3 . Write the matrices of the symmetries r t and j with respect to the standard basis of R 3 . Denote these matrices by R t and J , respectively. Confirm the relation JR t D R t J . 2.3.4. Consider the group D n of symmetries of the n -gon. (a) Show that 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 . (b) Show that the “flips” j k =n D r k2 =n j D r k j , for k 2 Z , are symmetries of the n -gon. (c) Show that the distinct flip symmetries of the
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Unformatted text preview: n-gon are r k j for k D 0;1;:::;n ± 1 . 2.3.5. (a) Show that 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 Figure 2.3.2 on page 107 for the case n D 5 . (b) If n is even and k is even, show that j k±=n D r k j is a rotation about an axis passing through a pair of opposite vertices of the n-gon. (c) Show that if n is even and k is odd, then j k±=n D r k j is a rotation about an axis passing through the midpoints of a pair of opposite edges of the n-gon. See Figure 2.3.2 on page 107 for the case n D 6 . 2.3.6. Find a subgroup of D 6 that is isomorphic to D 3 . 2.3.7. Find a subgroup of D 6 that is isomorphic to the symmetry group of the rectangle....
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