4023f11ps9

4023f11ps9 - j Orb x j = j G j = j Stab x j for each x 2...

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Homework #9 Due: November 18, 2011 1. For each of the following subsets of the plane, determine the symmetry group. Re- member that the possible symmetry groups are C n and D n for some n . See De±nition 5.2.4, the bottom of page 217 and the ±rst paragraph of page 218 for the description of C n and D n . (a) (b) (c) 2. If ± = ± 1 2 3 ² ± 4 5 ² then G = ( ± ), the cyclic subgroup of S 5 generated by ± , is a subgroup of S 5 of order 6. As a subgroup of S 5 , G acts on the set X = f 1 ; 2 ; 3 ; 4 ; 5 g . (a) Determine the orbit, Orb( x ) for each x 2 X . (b) Determine the stabilizer subgroup Stab( x ) for each x 2 X . (c) Use your results in part (a) to verify that j Orb( x ) j = j G j = j Stab( x ) j for each x 2 X . (This veri±es the result of Theorem 5.4.3 (Orbit-Stabilizer Theorem) in this case.) 3. If ± = ± 1 2 3 4 ² ± 5 6 ² then G = ( ± ), the cyclic subgroup of S 6 generated by ± , is a subgroup of S 6 of order 4. As a subgroup of S 6 , G acts on the set X = f 1 ; 2 ; 3 ; 4 ; 5 ; 6 g . (a) Determine the orbit, Orb( x ) for each x 2 X . (b) Determine the stabilizer subgroup Stab( x ) for each x 2 X . (c) Use your results in part (a) to verify that
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Unformatted text preview: j Orb( x ) j = j G j = j Stab( x ) j for each x 2 X . (This veri±es the result of Theorem 5.4.3 (Orbit-Stabilizer Theorem) in this case.) 4. (a) Find the number of di²erent squares with vertices colored red, white, or blue. (b) Find the number of di²erent m-colored squares for any m . 5. A wheel is divided evenly into six di²erent compartments. Each compartment can be painted red or white. The back of the wheel is black. How many di²erent color wheels are there? 6. A rectangular design consists of 10 parallel stripes of equal width. If each stripe can be painted red, blue, or green, ±nd the number of possible patterns. 7. Each side of an equilateral triangle is divided into two equal parts, and each part is colored red, yellow, or green. Find the number of patterns. Math 4023 1...
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This document was uploaded on 12/28/2011.

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