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Unformatted text preview: (a) Determine the orbit, Orb( x ) for x = 1 and x = 2. (b) Determine the stabilizer subgroup Stab( x ) for x = 1 and x = 2. (c) Use your results to verify that j Orb( x ) j = j G j = j Stab( x ) j for x = 1 and x = 2. 6. [9 Points] Fill in the missing parts to complete the statement of Burnside’s Theorem: If G is a ﬂnite group that acts on a ﬂnite set X , then the number k of distinct orbits is: k = 1 X : 7. [25 Points] Find the number of patterns obtained by coloring the vertices of a regular pentagon with 4 colors. Be sure to explain your calculations, including identiﬂcation of the symmetry group and all of the ﬂxed point calculations needed. Math 4023 November 21, 2011 1...
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This document was uploaded on 12/28/2011.
 Fall '09

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