Combinatorics Through Guided Discovery, 2004a

6.2. Groups Acting on Sets 115
We have seen that the fact that we have defined a permutation group as the
permutations of some specific set doesn’t preclude us from thinking of the elements
of that group as permuting the elements of some other set as well. In order to keep
track of which permutations of which set we are using to define our group and
which other set is being permuted as well, we introduce some new language and
notation. We are going to say that the group D 4 “acts” on the edges and diagonals
of a square and the group R of permutations of the vertices of a cube that arise from
rigid motions of the cube “acts” on the edges, faces, diagonals, etc. of the cube.
• Problem 280. In Figure 6.1.3 we show a cube with the positions of its vertices
and faces labeled. As with motions of the square, we let we let ϕ(x) be the
label of the place where vertex previously in position x is now.
(a) In Problem 263 we wrote in two row notation the permutation ρ of
the vertices that corresponds to rotating the cube 90 degrees around
a vertical axis through the faces t (for top) and u (for underneath).
(We rotated in a right-handed fashion around this axis, meaning that
vertex 6 goes to the back and vertex 8 comes to the front.) Write in two
row notation the permutation ρ of the faces that corresponds to this
member ρ of R.
(b) In Problem 263 we wrote in two row notation the permutation ϕ that
rotates the cube 120 degrees around the diagonal from vertex 1 to
vertex 7 and carries vertex 8 to vertex 6. Write in two row notation the
ϕ of the faces that corresponds to this member of R.
(c) In Problem 263 we computed the two row notation for ρ ◦ ϕ. Now
compute the two row notation for ρ ◦ ϕ (ρ was defined in Part 280.a),
and write in two row notation the permutation ρ ◦ ϕ of the faces that
corresponds to the action of the permutation ρ ◦ ϕ on the faces of the
cube. (For this question it helps to think geometrically about what
motion of the cube is carried out by ρ ◦ ϕ.) What do you observe
about ρ ◦ ϕ and ρ ◦ ϕ?
We say that a permutation group G acts on a set S if, for each member σ of G
there is a permutation σ of S such that
σ ◦ ϕ = σ ◦ ϕ
for every member σ and ϕ of G. In Problem 280.c you saw one example of this
condition. If we think intuitively of ρ and ϕ as motions in space, then following
the action of ϕ by the action of ρ does give us the action of ρ ◦ ϕ. We can also
reason directly with the permutations in the group R of rigid motions (rotations)
of the cube to show that R acts on the faces of the cube.
Problem 281. Show that a group G of permutations of a set S acts on S with
ϕ = ϕ for all ϕ in G.