§ Clean way to write burnside lemma
Burnside lemma says that . We prove this
- From orbit stabilizer, we know that .
- Since is the total cardinality of the orbit, each element in the orbit contributes towards cardinality of the full orbit.
- Thus, the sum over an orbit will be 1.
- Suppose a group action has two orbits, and . I can write the sum as: , which is equal to 2.
- I can equally write the sum as . But this sum is equal to .
- This sum sums over the entire group, so it can be written as .
- In general, the sum over the entire group will be the number of orbits, since the same argument holds for each orbit .
So we have derived:
If we have a transformation that fixes many things, ie, is large, then this is not helping "fuse" orbits of together, so the number of orbits will increase.