• Support PF! Buy your school textbooks, materials and every day products Here!

Application of Burnside's Theorem, Colorings

  • Thread starter saubbie
  • Start date
13
0
1. Homework Statement
I have multiple problems, all dealing with Burnside's Theorem, perhaps help on one would help explain the others. How many ways may the faces of a cube be colored using 3 colors, up to symmetry of the cube. And, how many ways may the faces of a dodecahedron be colored using 5 colors up to symmetry.


2. Homework Equations

Burnside's Theorem: # colorings= 1/(group of symmetry)*Sum of number of fixed colorings

3. The Attempt at a Solution

The symmetry groups are S4 and A5 respectively. The part that I get hung up on is the number of fixed colorings. I know for the identity permutation on the cube, all colorings are fixed, but I don't know how the other rotations work. Please help!
 

Answers and Replies

Related Threads for: Application of Burnside's Theorem, Colorings

  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
3
Views
11K
  • Last Post
Replies
0
Views
3K
  • Last Post
Replies
0
Views
4K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
1
Views
22K
Replies
8
Views
1K
Top