nLab transposition permutation

Idea

In group theory a transposition is a permutation that exchanges two elements and leaves all others fixed.

Definition

For $n \in \mathbb{N}$ and $1 \leq i \neq j \leq n$, the transposition of $i$ with $j$ among $\{1,\cdots, n\}$ is the permutation

$$\array{ \{1, \cdots, n\} & \underoverset{\simeq}{\;\;\; \sigma_{i,j} \;\;\; }{\longrightarrow} & \{1, \cdots, n\} \\ k &\mapsto& \left\{ \array{ k &\vert& k \notin \{i,j\} \\ j &\vert& k = i \\ i &\vert& k = j } \right. }$$

Properties

• Transpositions are generators for the symmetric group, exhibiting it as a finitely generated group.

• In fact already just the adjacent transpositions $t_{i , i+1}$ for $1 \leq i \leq n$ generate the symmetric group, see at Kendall tau distance.

Related concepts

• permutation representation, permutation matrix

• cyclic permutation, rotation permutation