Permutations
Definition
A permutation σ\sigma on a set XX is, to give three different phrasings,
a finite symmetry over XX
a bijection over XX with finite support
a bijection over XX 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 σ\sigma over a set XX by juxtaposing the elements x1,,xnx_1, \dots, x_n of XX with their respective images σ(x1),,σ(xn)\sigma(x_1), \dots, \sigma(x_n). For instance, we might write a bijection over {1,,7}\{1, \dots, 7\} as:
σ=(12345674531762)\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 XX has an understood ordering, we may omit the first line from two-line notation to produce: σ=(4531762)\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: σ=4531762\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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