Combinatorics Through Guided Discovery, 2004a

6.2. Groups Acting on Sets 121
Problem 292.
(a) If G is a group, how is the set {τσ | τ ∈ G} related to G?
(b) Use this to show that y is in the multiorbit Gx multi if and only if
Gx multi = Gy multi .
Problem 292.b tells us that, when G acts on S, each element x of S is in one and
only one multiorbit. Since each orbit is a subset of a multiorbit and each element x
in S is in one and only one orbit, this also tells us there is a bijection between the
orbits of G and the multiorbits of G, so that we have the same number of orbits as
multiorbits.
When a group acts on a set, a group element is said to fix an element of x ∈ S if
σ(x) =x. The set of all elements fixing an element x is denoted by (x).
Problem 293. Suppose a group G acts on a set S. What is special about the
subset (x) for an element x of S?
• Problem 294. Suppose a group G acts on a set S. What is the relationship
of the multiplicity of x ∈ S in its multiorbit and the size of (x)?
Problem 295. What can you say about relationships between the multiplicity
of an element y in the multiorbit Gx multi and the multiplicites of other
elements? Try to use this to get a relationship between the size of an orbit
of G and the size of G. (h)
We suggested earlier that a quotient principle for multisets might prove useful.
The quotient principle came from the sum principle, and we do not have a sum
principle for multisets. Such a principle would say that the size of a union of disjoint
multisets is the sum of their sizes. We have not yet defined the union of multisets
or disjoint multisets, because we haven’t needed the ideas until now. We define the
union of two multisets S and T to be the multiset in which the multiplicity of an
element x is the maximum5 of the multiplicity of x in S and its multiplicity in T .
Similarly, the union of a family of multisets is defined by defining the multiplicity of
an element x to be the maximum of its multiplicities in the members of the family.
Two multisets are said to be disjoint if no element is a member of both, that is, if
no element has multiplicity one or more in both. Since the size of a multiset is the
sum of the multiplicities of its members, we immediately get the sum principle for
multisets.
The size of a union of disjoint multisets is the sum of their sizes.
5We choose the maximum rather than the sum so that the union of sets is a special case of the union
of multisets.