hw3 - formulas for R i R j R i F j F i R j and F i F j For...

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Math 113 Homework # 3, due 9/23/9 at 2:10 PM 1. Fraleigh section 4, exercise 41. 2. Fraleigh section 5, exercise 13. 3. (a) Fraleigh section 5, exercise 54. (b) Is this still true if one replaces intersection by union? Prove or give a counterexample. 4. Let n > 1 be an integer and let θ = 2 π/n . Let P be the regular n -gon with vertices (cos iθ, sin ) for i Z n . The dihedral group D n is the symmetry group of P , which consists of rotations R i and reflections F i for i Z n . Here R i is the counterclockwise rotation around the origin by angle , and F i is the reflection across the line through the origin and (cos iθ/ 2 , sin iθ/ 2). Your problem: find (and give at least some justification for) general
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Unformatted text preview: formulas for R i R j , R i F j , F i R j , and F i F j . For example, R i R j = R i + j , where the addition of indices is mod n . 5. Find all subgroups of D 4 . 6. If G is a group, the center of G is deﬁned to be Z ( G ) = { x ∈ G | xy = yx for all y ∈ G } . (a) Show that Z ( G ) is a subgroup of G . (b) For n > 2, what is the center of D n ? (Use the multiplication rules you found above. The answer depends on whether n is even or odd.) 7. Fraleigh section 6, exercise 32 (justify as always). 8. How challenging did you ﬁnd this assignment? How long did it take?...
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