Combinatorics Through Guided Discovery, 2004a

118 6. Groups acting on sets
3, and so on. Then a coloring of the edges with 12 red, 23 blue, 34 red and
41 blue can be represented as
{(12, R), (23, B), (34, R), (41, B)} . (6.4)
If ρ is the rotation through 90 degrees, then we have a permutation ρ acting
on its edges. This permutation acts on the colorings to give us a permutation
ρ of the set of colorings.
(a) What is ρ of the coloring in (6.4)?
(b) What is ρ 2 of the coloring in (6.4)?
If G is a group that acts on the set S, we define the action of G on the colorings
(S, f ) by by
σ((S, f )) = σ ({ (x, f (x)) | x ∈ S }) = { (σ(x), f (x)) | x ∈ S } .. (6.5)
We have two bars over σ because σ is a permutation of one set that gives us a
permutation σ of a second set, and then σ acts to give a permutation σ of a thid set,
the set of colorings. For example, suppose we want to analyze colorings of the faces
of a cube under the action of the rotation group of the cube as we have defined it
on the vertices. Each vertex-permutation σ in the group gives a permutation σ of
the faces of the cube. Then each permutation σ of the faces gives us a permutation
σ of the colorings of the faces.
In the special case that G is a group of permutations of S rather than a group
acting on S, Equation (6.5) becomes
σ((S, f )) = σ({(x, f (x)) | x ∈ S}) ={(σ(x), f (x)) | x ∈ S}.
In the case where G is the rotation group of the square acting on the vertices of the
square, the example of acting on a coloring by ρ that we saw in (6.3) is an example
of this kind of action. In the standard notation, when we act on a coloring by σ,the
color in position i moves to position σ(i).
Problem 285. Why does the action we have defined on colorings in Equation
(6.5) take a coloring to a coloring?
Problem 286. Show that if G is a group of permutations of a set S, and f is
a coloring function on S, then the equation
σ({(x, f (x)) | x ∈ S}) ={(σ(x), f (x)) | x ∈ S}
defines an action of G on the colorings (S, f ) of S. (h)