Combinatorics Through Guided Discovery, 2004a

6.1. Permutation Groups 107
and σ(4) = 2, we write
( )
1 2 3 4
σ =
.
3 1 4 2
We call this notation the two row notation for permutations. In the two row
notation for a permutation of {a 1 , a 2 ,...,a n }, we write the numbers a 1 through a n
in a one row and we write σ(a 1 ) through σ(a n ) in a row right below, enclosing both
rows in parentheses. Notice that
( )
1 2 3 4
=
3 1 4 2
( )
2 1 4 3
,
1 3 2 4
although the second ordering of the columns is rarely used.
If ϕ is given by
( )
1 2 3 4
ϕ =
,
4 1 2 3
then, by applying the definition of composition of functions, we may compute σ ◦ ϕ
as shown in Figure 6.1.2.
1 2 3 4
( ) (
4 1 2 3
)
= 1 2 3 4
(
2 3 1 4
)
1 2 3 4
3 1 4 2
Figure 6.1.2: How to multiply permutations in two-row notation.
We don’t normally put the circle between two permutations in two row notation
when we are composing them, and refer to the operation as multiplying the permutations,
or as the product of the permutations. To see how Figure 6.1.2 illustrates
composition, notice that the arrow starting at 1 in ϕ goes to 4. Then from the 4 in
ϕ it goes to the 4 in σ and then to 2. This illustrates that ϕ(1) = 4 and σ(4) = 2, so
that σ(ϕ(1)) = 2.
( )( )
1 2 3 4 5 1 2 3 4 5
Problem 258. For practice, compute
.
3 4 1 5 2 4 3 5 1 2
6.1.4 The dihedral group
We found four permutations that correspond to rotations of the square. In Problem
255 you found four permutations that correspond to flips of the square in
space. One flip fixes the vertices in the places labeled 1 and 3 and interchanges
the vertices in the places labeled 2 and 4. Let us denote it by ϕ 1|3 . One flip fixes
the vertices in the positions labeled 2 and 4 and interchanges those in the positions
labeled 1 and 3. Let us denote it by ϕ2|4. One flip interchanges the vertices in the
places labeled 1 and 2 and also interchanges those in the places labeled 3 and 4.