Editing Permutation group (section)

==Notation==
{{main|Permutation#Notations}}
Since permutations are [[bijection]]s of a set, they can be represented by [[Augustin-Louis Cauchy|Cauchy]]'s ''two-line notation''.<ref>{{citation|title=The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory|first=Hans|last=Wussing|publisher=Courier Dover Publications|year=2007|isbn=9780486458687|page=94|url=https://books.google.com/books?id=Xp3JymnfAq4C&pg=PA94|quote=Cauchy used his permutation notation—in which the arrangements are written one below the other and both are enclosed in parentheses—for the first time in 1815.}}</ref> This notation lists each of the elements of ''M'' in the first row, and for each element, its image under the permutation below it in the second row. If <math>\sigma</math> is a permutation of the set <math>M = \{x_1,x_2,\ldots,x_n\}</math> then,
: <math> \sigma = \begin{pmatrix}
x_1 & x_2 & x_3 & \cdots & x_n \\
\sigma(x_1) &\sigma(x_2) & \sigma(x_3) & \cdots& \sigma(x_n)\end{pmatrix}.</math>
For instance, a particular permutation of the set {1, 2, 3, 4, 5} can be written as
: <math>\sigma=\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
2 & 5 & 4 & 3 & 1\end{pmatrix};</math>
this means that ''σ'' satisfies ''σ''(1) = 2, ''σ''(2) = 5, ''σ''(3) = 4, ''σ''(4) = 3, and ''σ''(5) = 1. The elements of ''M'' need not appear in any special order in the first row, so the same permutation could also be written as
: <math>\sigma=\begin{pmatrix}
3 & 2 & 5 & 1 & 4 \\
4 & 5 & 1 & 2 & 3\end{pmatrix}.</math>

Permutations are also often written in [[cycle notation]] (''cyclic form'')<ref>especially when the algebraic properties of the permutation are of interest.</ref> so that given the set ''M'' = {1, 2, 3, 4}, a permutation ''g'' of ''M'' with ''g''(1) = 2, ''g''(2) = 4, ''g''(4) = 1 and ''g''(3) = 3 will be written as (1, 2, 4)(3), or more commonly, (1, 2, 4) since 3 is left unchanged; if the objects are denoted by single letters or digits, commas and spaces can also be dispensed with, and we have a notation such as (124). The permutation written above in 2-line notation would be written in cycle notation as <math> \sigma = (125)(34).</math>