Adv Alegbra HW Solutions 81

Adv Alegbra HW Solutions 81 - 81 2.140 Let X be a disk...

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81 2.140 Let X be a disk divided into n congruent circular sectors, and let ρ be a rotation by ( 360 / n ) .D e f ne a roulette wheel to be a ( q , G ) -coloring, where the cyclic group G =h ρ i of order n is acting. Prove that if n = 6, then there are 1 6 ( 2 q + 2 q 2 + q 3 + q 6 ) roulette wheels having 6 sectors. Solution. The group here is G =h ρ i of order 6 acting by rotations. Thus, ρ = ( 123456 ), ρ 2 = ( 135 )( 246 ), ρ 3 = ( 14 )( 25 )( 36 ), ρ 4 = ( 153 )( 24 ), ρ 5 = ( 165432 ). Hence P G ( x 1 ,..., x 6 ) = 1 6 ( x 6 1 + 2 x 6 + 2 x 2 3 + x 3 2 ) and P G ( q ,..., q ) = 1 6 ( q 6 + 2 q + 2 q 2 + q 3 ). 2.141 Let X be the vertices of a regular n -gon, and let the dihedral group G = D 2 n act (as the usual group of symmetries). De f ne a bracelet to be a ( q , G ) -coloring of a regular n -gon, and call each of its vertices a bead . (i) How many bracelets are there having 5 beads, each of which can be colored any one of q available colors? Solution. Proceed for the pentagon as we did for the square in Example 2.139. If { v 0 ,v 1 ,v 2 ,v 3
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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