hw3 - r i r j , r i s j , s i r j , and s i s j . For...

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Math 113 Homework 3, due 2/11/10 at 9:30 AM Policy: if you worked with other people on this assignment, write their names on the front of your homework. Remember that you must write up your solutions independently. 1. Fraleigh, section 6, exercise 32 (as always, carefully justify your answers). 2. Fraleigh, section 6, exercises 44 and 56(a). 3. Let n 2 be an integer, and let θ = 2 π/n . Consider the regular n -gon P with vertices (cos kθ, sin ) for k Z n . The dihedral group D n is the set of symmetries of P , which consists of rotations r j and re±ections s j for j Z n . Here r j is the counterclockwise rotation around the origin by angle , and s j is the re±ection across the line through the origin and (cos jθ/ 2 , sin jθ/ 2). The binary operation is composition. (a) Find (and give at least some justi²cation for) general formulas for the products
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Unformatted text preview: r i r j , r i s j , s i r j , and s i s j . For example, r i r j = r i + j , where the addition of indices is mod n . (b) Show that D n is indeed a group. 4. Find all subgroups of D 5 . (Hint: rst prove that, if H D 5 contains a non-trivial rotation, then it contains all rotations.) 5. If G is a group, the center of G is dened 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 3, what is the center of D n ? (Use the multiplication rules you found above. The answer depends on whether n is even or odd.) 6. How challenging did you nd this assignment? How long did it take?...
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This note was uploaded on 03/14/2012 for the course MATH 113 taught by Professor Ogus during the Spring '08 term at University of California, Berkeley.

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