16.3 Transpositions Generate Permutations

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 16.2 Symmetric Groups

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Transpositions Generate Permutations

All cycles can be written as a composition of transpositions. In other words, transpositions generate $S_{n}$ .

Let's look back at

S_{3}

Let's look at a symmetric group of 3 elements.
$S_{3} = {π : π [3] \to [3]}$

Recall the 3-cycle of $S_{3}$ .

We can rewrite this as a composition of two transpositions.

We can write this in function notation as

\begin{aligned} π_{5} & = π_{2} \circ π_{3} \\ π_{5} (1) & = π_{3} (π_{2} (1)) \\ = π_{3} (2) \\ = 2 \end{aligned}