Combinatorics Through Guided Discovery, 2004a

120 6. Groups acting on sets
Problem 290. Complete the proof of Theorem 6.2.3.
Notice that thinking in terms of orbits actually hides some information about
the action of our group. When we computed the multiset of all results of acting on
{1, 2} with the elements of D 4 , we got an eight-element multiset containing each
side twice. When we computed the multiset of all results of acting on {1, 3} with
the elements of D 4 , we got an eight-element multiset containing each diagonal of
the square four times. These multisets remind us that we are acting on our twoelement
sets with an eight-element group. The multiorbit of G determined by an
element x of S is the multiset
{σ(x) | σ ∈ G},
and is denoted by Gx multi .
When we used the quotient principle to count circular seating arrangements or
necklaces, we partitioned up a set of lists of people or beads into blocks of equivalent
lists. In the case of seating n people around a round table, what made two lists
equivalent was, in retrospect, the action of the rotation group C n . In the case of
stringing n beads on a string to make a necklace, what made two lists equivalent
was the action of the dihedral group. Thus the blocks of our partitions were orbits
of the rotation group or the dihedral group, and we were counting the number of
orbits of the group action. In Problem 45, we were not able to apply the quotient
principle because we had blocks of different sizes. However, these blocks were still
orbits of the action of the group D 4 . And, even though the orbits have different
sizes, we expect that each orbit corresponds naturally to a multiorbit and that the
multiorbits all have the same size. Thus if we had a version of the quotient rule for
a union of multisets, we could hope to use it to count the number of multiorbits.
Problem 291.
(a) Find the orbit and multiorbit of D 4 acting on the coloring
{(1, R), (2, R), (3, B), (4, B)},
or, in standard notation, RRBB of the vertices of a square.
(b) How many group elements map the coloring RRBB to itself? What is
the multiplicity of RRBB in its multiorbit?
(c) Find the orbit and multiorbit of D 4 acting on the coloring
{(1, R), (2, B), (3, R), (4, B)}.
(d) How many elements of the group send the coloring RBRB to itself?
What is the multiplicity of RBRB in its orbit?