nLab
transposition permutation

Contents

Contents

Idea

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

Definition

For nn \in \mathbb{N} and 1ijn1 \leq i \neq j \leq n, the transposition of ii with jj among {1,,n}\{1,\cdots, n\} is the permutation

{1,,n} σ i,j {1,,n} k {k | k{i,j} j | k=i i | k=j \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

Created on April 17, 2021 at 11:49:29. See the history of this page for a list of all contributions to it.