Group theory and generalizations Definition

Transposition

Builds on Symmetric group Cycle notation

Definition

Let $X$ be a set and let $a, b \in X$ with $a \neq b$ . The transposition $(a\ b)$ is the permutation of $X$ that sends $a$ to $b$ , sends $b$ to $a$ , and fixes each other element of $X$ .

Remarks

A transposition is a $2$ -cycle in cycle notation, and $(a\ b) = (b\ a)$ . Each transposition is its own inverse, so it has order $2$ . The symmetric group $S_n$ contains exactly $\binom{n}{2}$ transpositions.

Reading path

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Cycle notation concept
Group concept
Symmetric group definition
Transposition definition

Where this leads

18 pages build on this one.

Tag	`02DTRYKGE8`
URI	`m2://02DTRYKGE8`
Aliases	`two-cycle`
Kind	definition
Writing	W0 stub
MSC	20B30
Defines	Transposition
Uses	Symmetric group, Cycle notation

Transposition

Definition

Remarks

Reading path

Where this leads

Further reading

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