This preview shows page 1. Sign up to view the full content.
Unformatted text preview: 9 432 arrangements of the beads. Lets 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 nite group G on a nite set X , consider the set F D f .g;x/ 2 G 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 :...
View
Full
Document
This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Counting

Click to edit the document details