Combinatorics Through Guided Discovery, 2004a

110 6. Groups acting on sets
Problem 265. As with a two-dimensional figure, it is possible to talk about
isometries of a three-dimensional figure. These are distance preserving
functions from the figure to itself. The function that reflects the cube in
Figure 6.1.3 through a plane halfway between the bottom face and top face
exchanges the vertices 1 and 5, 2 and 6, 3 and 7, and 4 and 8 of the cube. This
function preserves distances between points in the cube. However, it cannot
be achieved by a rigid motion of the cube because a rigid motion that takes
vertex 1 to vertex 5, vertex 2 to vertex 6, vertex 3 to vertex 7, and vertex 4 to
vertex 8 would not return the cube to its original location; rather it would
put the bottom of the cube where its top previously was and would put the
rest of the cube above that square rather than below it.
(a) How many elements are there in the group of permutations of [8] that
correspond to isometries of the cube? (h)
(b) Is every permutation of [8] that corresponds to an isometry either a
rotation or a reflection? (h)
6.1.5 Group tables (Optional)
We can always figure out the composition of two permutations of the same set
by using the definition of composition, but if we are going to work with a given
permutation group again and again, it is worth making the computations once
and recording them in a table. For example the group of rotations of the square
may be represented as in Table 6.1.4. We list the elements of our group, with the
identity first, across the top of the table and down the left side of the table, using
the same order both times. Then in the row labeled by the group element σ and the
column labelled by the group element ϕ, we write the composition σ ◦ ϕ, expressed
in terms of the elements we have listed on the top and on the left side. Since a
group of permutations is closed under composition, the result σ ◦ ϕ will always be
expressible as one of these elements.
◦ ι ρ ρ 2 ρ 3
ι ι ρ ρ 2 ρ 3
ρ ρ ρ 2 ρ 3 ι
ρ 2 ρ 2 ρ 3 ι ρ
ρ 3 ρ 3 ι ρ ρ 2
Table 6.1.4: The group table for the rotations of a square.
Problem 266. In Table 6.1.4, all the entries in a row (not including the first
entry, the one to the left of the line) are different. Will this be true in any
group table for a permutation group? Why or why not? Also in Table 6.1.4
all the entries in a column (not including the first entry, the one above the