College Algebra Exam Review 237

College Algebra Exam Review 237 - colors. Count the number...

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5.3. SYMMETRIES OF GROUPS 247 It is required to count the orbits for the action of the rotation group G of the cube on the set X of 3 6 colorings of the faces of the cube. For each g 2 G the number of x 2 X that are fixed by g is 3 N.g/ , where N.g/ is the number of orbits of h g i acting on faces of the cube. We compute the following data: n -fold rotation axis, order of N.g/ number of such n D rotation group elements ± 1 6 1 2 2 3 6 3 3 2 8 4 4 3 6 4 2 4 3 Thus, the number of colorings of the faces of the cube with three colors is 1 j G j X g 2 G j Fix .g/ j D 1 24 .3 6 C 8 ² 3 2 C 6 ² 3 3 C 3 ² 3 4 C 6 ² 3 3 / D 57: Exercises 5.2 5.2.1. How many necklaces can be made with six beads of three different colors? 5.2.2. How many necklaces can be made with two red beads, two green beads, and two violet beads? 5.2.3. Count the number of ways to color the edges of a cube with four
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Unformatted text preview: colors. Count the number of ways to color the edges of a cube with r colors; the answer is a polynomial in r . 5.2.4. Count the number of ways to color the vertices of a cube with three colors. Count the number of ways to color the vertices of a cube with r colors. 5.2.5. Count the number of ways to color the faces of a dodecahedron with three colors. Count the number of ways to color the faces of a dodecahe-dron with r colors. 5.3. Symmetries of Groups A mathematical object is a set with some structure. A bijection of the set that preserves the structure is undetectable insofar as that structure is...
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