Combinatorics Through Guided Discovery, 2004a

116 6. Groups acting on sets
• Problem 282. The group D 4 is a group of permutations of {1, 2, 3, 4} as
in Problem 255. We are going to show in this problem how this group
acts on the two-element subsets of {1, 2, 3, 4}. In Problem 287 we will see
a natural geometric interpretation of this action. In particular, for each
two-element subset {i, j} of {1, 2, 3, 4} and each member σ of D 4 we define
σ({i, j}) ={σ(i),σ(j)}. Show that with this definition of σ, the group D 4
acts on the two-element subsets of {1, 2, 3, 4}.
• Problem 283. Suppose that σ and ϕ are permutations in the group R of
rigid motions of the cube. We have argued already that each rigid motion
sends a face to a face. Thus σ and ϕ both send the vertices on one face to
the vertices on another face. Let {h, i, j, k} be the set of labels next to the
vertices on a face F.
(a) What are the vertices of the face F ′ that F is sent to by ϕ?
(b) What are the vertices of the face F ′′ that F ′ is sent to by σ?
(c) What are the vertices of the face F ′′′ that F is sent to by σ ◦ ϕ?
(d) How have you just shown that the group R acts on the faces?
6.2.1 Groups acting on colorings of sets
Recall that when you were asked in Problem 45 to find the number of ways to
place two red beads and two blue beads at the corners of a square free to move
in three-dimensional space, you were not able to apply the quotient principle to
answer the question. Instead you had to see that you could divide the set of six
lists of two Rs and two Bs into two sets, one of size two in which the Rs and Bs
alternated and one of size four in which the two reds (and therefore the two blues)
would be side-by-side on the square. Saying that the square is free to move in space
is equivalent to saying that two arrangements of beads on the square are equivalent
if a member of the dihedral group carries one arrangement to the other. Thus an
important ingredient in the analysis of such problems will be how a group can
act on colorings of a set of vertices. We can describe the coloring of the square in
Figure 6.2.2 as the function f with
f (1) = R, f (2) = R, f (3) = B, and f (4) = B,
but it is more compact and turns out to be more suggestive to represent the coloring
in Figure 6.2.2 as the set of ordered pairs
(1, R), (2, R), (3, B), (4, B) (6.1)