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group theory

Contents

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

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