Permutations

Definition

A permutation $\sigma$ on a set $X$ is, to give three different phrasings,
Permutations are typically considered as elements of symmetric groups

Notation + Terminology

$\sigma = \begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7
\\ 4 & 5 & 3 & 1 & 7 & 6 & 2
\end{pmatrix}$

One-line notation more

In the case where $X$ has an understood ordering, we may omit the first line from two-line notation to produce:
$\sigma = \begin{pmatrix}
4 & 5 & 3 & 1 & 7 & 6 & 2
\end{pmatrix}$
In order to distinguish this from cycle notation, one-line notation is often written without parentheses:
$\sigma = \begin{matrix}
4 & 5 & 3 & 1 & 7 & 6 & 2
\end{matrix}$
I generally try to prefer two-line notation over one-line

Properties

All permutations can be decomposed into a product of disjoint cycles

Referenced by: