Combinatorics Through Guided Discovery, 2004a

108 6. Groups acting on sets
Let us denote it by ϕ 12|34 . The fourth flip interchanges the vertices in the places
labeled 1 and 4 and interchanges those in the places labeled 2 and 3. Let us denote
it by ϕ 14|23 . Notice that ϕ 1|3 is a permutation that takes the vertex in place 1 to the
vertex in place 1 and the vertex in place 3 to the vertex in place 3, while ϕ 12|34 is a
permutation that takes the edge between places 1 and 2 to the edge between places
2 and 1 (which is the same edge) and takes the edge between places 3 and 4 to the
edge between places 4 and 3 (which is the same edge). This should help to explain
the similarity in the notation for the two different kinds of flips.
• Problem 259. Write down the two-row notation for ρ 3 , ϕ 2|4 , ϕ 12|34 and
ϕ 2|4 ◦ ϕ 12|34 . Remember that σ(i) stands for the position where the vertex
that originated in position i is after we apply σ.
Problem 260. (You may have already done this problem in Problem 255,in
which case you need not do it again!) In Problem 255, if a rigid motion
of three-dimensional space returns the square to its original position, in
how many places can vertex number one land? Once the location of vertex
number one is decided, how many possible locations are there for vertex
two? Once the locations of vertex one and vertex two are decided, how
many locations are there for vertex three? Answer the same question for
vertex four. What does this say about the relationship between the four
rotations and four flips described above and the permutations you described
in Problem 255?
The four rotations and four flips of the square described before Problem 259
form a group called the dihedral group of the square. Sometimes the group is
denoted D 8 because it has eight elements, and sometimes the group is denoted by
D 4 because it deals with four vertices! Let us agree to use the notation D 4 for the
dihedral group of the square. There is a similar dihedral group, denoted by D n ,of
all the rigid motions of three-dimensional space that return a regular n-gon to its
original position (but might put the vertices in different places.)
Problem 261. Another view of the dihedral group of the square is that it
is the group of all distance preserving functions, also called isometries,
from a square to itself. Notice that an isometry must be a bijection. Any
rigid motion of the square preserves the distances between all points of the
square. However, it is conceivable that there might be some isometries that
do not arise from rigid motions. (We will see some later on in the case of
a cube.) Show that there are exactly eight isometires (distance preserving
functions) from a square to itself. (h)