2 Sets and functions 2.15 Powers and orders 2.17 Sign

2.16 Transpositions

2.16.1 Definition of a transposition

Definition 2.16.1.

A transposition is a 2-cycle.

For example, the only transpositions in $S_{3}$ are $(1,2)$ , $(2,3)$ , and $(1,3)$ .

2.16.2 Every permutation is a product of transpositions

In this section we’re going to prove that every permutation can be written as a product of transpositions. Before we do so, here is some motivation for why we should expect this to be true.

One way we’ve already seen of illustrating a permutation $s$ in $S_{n}$ is to draw the numbers $1,2,\ldots,n$ in a column, and then in another column to its right, and draw a line from each number $i$ in the first column to $s(i)$ in the second. You get what looks like a lot of pieces of string tangled together.

Here is a diagram of the permutation $s=(1,2,3,4,5)$ . Think of the lines as pieces of string connecting $i$ on the left with $s(i)$ on the right.

Figure 2.17: A string diagram of the permutation

(1,2,3,4,5)

. Two columns contain the numbers 1, 2, 3, 4, 5. Strings connect the numbers 1, 2, 3, 4, 5 in the left-hand column to 2, 3, 4, 5, 1 respectively in the right-hand column.

Composing two permutations drawn in this way corresponds to placing their diagrams side-by-side:

Figure 2.18: String diagrams for

(1,2)

and

(2,3)

are shown side by side, and then the right-hand column of the diagram for

(1,2)

is joined to the left-hand column for

(2,3)

. The resulting diagram represents

(2,3)(1,2)=(1,3,2)

Imagine taking such a diagram and stretching it out. You could divide it up into smaller diagrams, each of which contains only one crossing of strings.

Figure 2.19: String diagram for

(1,2,3,4)

, with dotted vertical lines to divide the strings into sections with only one crossing. From left to right, the first crossing is between strings 3 and 4, then 2 and 3, then 1 and 2

A diagram with a single string crossing is a transposition, since only two numbers change place. The diagram above illustrates the fact that

(1,2,3,4)=(1,2)(2,3)(3,4).

Now we’re ready for a formal proof of the result that every permutation equals a product of transpositions. We first do it for cycles:

Lemma 2.16.1.

Every cycle equals a product of transpositions.

Proof.

Let $a=(a_{0},\ldots,a_{m-1})$ be a cycle. Lemma 2.14.1 tells us that

a=(a_{0},a_{1})(a_{1},a_{2},\ldots,a_{m-1}).

Using that lemma again on the cycle $(a_{1},\ldots,a_{m-1})$ we get

a=(a_{0},a_{1})(a_{1},a_{2})(a_{2},a_{3},\ldots,a_{m-1}).

Repeating this gives

a=(a_{0},a_{1})(a_{1},a_{2})\cdots(a_{m-2},a_{m-1})

which shows that $a$ can be written as a product of transpositions. ∎

To illustrate this result you should check by computing both sides that

(1,2,3,4)=(1,2)(2,3)(3,4).

Theorem 2.16.2.

Every permutation in $S_{n}$ is equal to a product of transpositions.

Proof.

Let $p$ be a permutation. We have seen that every permutation can be written as a product of cycles, so there are cycles $c_{1},\ldots,c_{k}$ such that $p=c_{1}\cdots c_{k}$ . The lemma above shows how to write each $c_{i}$ as a product of transpositions, which expresses $p$ as a product of transpositions too. ∎

Example 2.16.1.

Suppose we want to express

s=\begin{pmatrix}1&2&3&4&5&6\\ 2&3&1&5&6&4\end{pmatrix}

as a product of transpositions. A disjoint cycle decomposition for $s$ is

s=(1,2,3)(4,5,6)

and applying the lemma above, we get

	$\displaystyle(1,2,3)$	$\displaystyle=(1,2)(2,3)$
	$\displaystyle(4,5,6)$	$\displaystyle=(4,5)(5,6)$

s=(1,2,3)(4,5,6)=(1,2)(2,3)(4,5)(5,6).