Permutations
Definition
A permutation $\sigma$ on a set $X$ is, to give three different phrasings,
a finite symmetry over $X$
a bijection over $X$ with finite support
a bijection over $X$ which is equivalent to the identity function on all but a finite number of elements
Permutations are typically considered as elements of symmetric groups
Notation + Terminology
Two-line notation: we can write a permutation $\sigma$ over a set $X$ by juxtaposing the elements $x_1, \dots, x_n$ of $X$ with their respective images $\sigma(x_1), \dots, \sigma(x_n)$. For instance, we might write a bijection over $\{1, \dots, 7\}$ as:
$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 4 & 5 & 3 & 1 & 7 & 6 & 2 \end{pmatrix}$
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

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