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Unformatted text preview: Name: Solutions Exam 3 Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without supporting work. Put your name on each page of your paper. 1. [30 Points] (a) Fill in the missing parts to complete the statement of Burnsides Theorem: If u1D43A is a finite group that acts on a finite set u1D44B , then the number u1D441 of distinct orbits is: u1D441 = 1 u1D43A uni2211.alt02 u1D454 u1D43A u1D44B u1D454 . (b) A wheel is divided evenly into 6 different wedges. Each wedge can be painted red (R), white (W), blue (B) or green (G). The back of the wheel is black. We consider two color patterns to be the same (or equivalent) provided one can be obtained from the other by means of a rotation about the center of the wheel. i. What are all of the rotations that make up the symmetry group of this wheel divided into 6 equal parts? List them in the form u1D70C u1D451 where u1D70C u1D451 denotes a coun- terclockwise rotation of degree u1D451 about the center of the wheel. Solution. The rotations that preserve the wedges are rotations u1D70C u1D451 where u1D451 is a multiple of 60 . Hence, there are 6 such rotations: u1D70C , u1D70C 60 , u1D70C 120 , u1D70C 180 , u1D70C 240 , and u1D70C 300 . ii. The following picture describes one color pattern. Draw all of the color patterns that are equivalent to this one. That is, the ones that are obtained from this one by means of all of the rotations. G W W R B R Solution. The following are the patterns obtained from the six possible rota- tions: B W G R W R R B W G R W W R B W G R R W R B W G G R W R B W W G R W R B u1D70C u1D70C 60 u1D70C 120 u1D70C 180 u1D70C 240 u1D70C...
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This document was uploaded on 12/28/2011.
- Fall '09