Combinatorics Through Guided Discovery, 2004a

122 6. Groups acting on sets
Taking the multisets all to have the same size, we get the product principle for
multisets.
The union of a set of m disjoint multisets, each of size n has size mn.
The quotient principle for multisets then follows immediately.
If a p-element multiset is a union of q disjoint multisets, each of size r,
then q = p/r.
• Problem 296. How does the size of the union of the set of multiorbits of a
group G acting on a set S relate to the number of multiorbits and the size of
G?
• Problem 297. How does the size of the union of the set of multiorbits of a
group G acting on a set S relate to the numbers | (x)|?
• Problem 298. In Problems 296 and 297 you computed the size of the union
of the set of multiorbits of a group G acting on a set S in two different ways,
getting two different expressions must be equal. Write the equation that
says they are equal and solve for the number of multorbits, and therefore
the number of orbits.
6.2.3 The Cauchy-Frobenius-Burnside Theorem
• Problem 299. In Problem 298 you stated and proved a theorem that expresses
the number of orbits in terms of the number of group elements
fixing each element of S. It is often easier to find the number of elements
fixed by a given group element than to find the number of group elements
fixing an element of S.
(a) For this purpose, how does the sum ∑ x : x∈S | (x)| relate to the
number of ordered pairs (σ, x) (with σ ∈ G and x ∈ S) such that σ
fixes x?
(b) Let χ(σ) denote the number of elements of S fixed by σ. How can the
number of ordered pairs (σ, x) (with σ ∈ G and x ∈ S) such that σ
fixes x be computed from χ(G)? (It is ok to have a summation in your
answer.)
(c) What does this tell you about the number of orbits?