Definition:Transposition

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Definition

Let $S$ be a set.

A transposition on $S$ is a $2$-cycle.

That is, a transposition is a permutation $\rho$ on $S$ which exchanges, or transposes, exactly two elements of $S$.


Thus if $\rho$ is a transposition which transposes two elements $r, s \in S$, it follows from the definition of fixed elements that:

$\Fix \rho = S \setminus \set {r, s}$


Adjacent Transposition

Let $S_n$ denote the symmetric group on $n$ letters.

An adjacent transposition is a transposition that exchanges two consecutive integers $j$ and $j + 1$, where $1 \le j < n$.

In cycle notation, they are denoted:

$\begin {pmatrix} j & j + 1 \end {pmatrix}$


Also known as

A transposition is colloquially known as a two-letter swap.


Sources

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