Unformatted text preview: 9Š 4Š3Š2Š arrangements of the beads. Let’s consider a simpler example that we can work out by inspection: Example 5.2.1. Consider necklaces made of two blue and two white beads. There are six arrangements of the beads at the vertices of the square, but only two orbits under the action of the dihedral group D 4 , namely, that with two blue beads adjacent and that with the two blue beads at opposite corners. One orbit contains four arrangements and the other two arrangements. We see from this example that the orbits will have different sizes, so we cannot expect the answer to the problem simply to be some divisor of the number of arrangements of beads. In order to count orbits for the action of a ﬁnite group G on a ﬁnite set X , consider the set F D f .g;x/ 2 G ² X W gx D x g . For g 2 G , let Fix .g/ D f x 2 X W gx D x g , and let 1 F .g;x/ D ( 1 if .g;x/ 2 F otherwise :...
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 Fall '08
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 Algebra, Counting, white beads

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