Editing Permutation (section)

===Composition of permutations===
There are two ways to denote the composition of two permutations. In the most common notation, <math>\sigma\cdot \tau</math> is the function that maps any element ''x'' to <math>\sigma(\tau(x))</math>. The rightmost permutation is applied to the argument first,<ref>
{{cite book | last1=Biggs | first1=Norman L. | last2=White | first2=A. T.
|year=1979
|publisher=Cambridge University Press
|title=Permutation groups and combinatorial structures
|isbn=978-0-521-22287-7
}}
</ref>
because the argument is written to the right of the function.

A ''different'' rule for multiplying permutations comes from writing the argument to the left of the function, so that the leftmost permutation acts first.<ref>
{{cite book |last1=Dixon |first1=John D. |url=https://archive.org/details/permutationgroup0000dixo |title=Permutation Groups |last2=Mortimer |first2=Brian |publisher=Springer |year=1996 |isbn=978-0-387-94599-6 |url-access=registration}}
</ref><ref>
{{cite book |last1=Cameron |first1=Peter J. |url=https://archive.org/details/permutationgroup0000came |title=Permutation groups |publisher=Cambridge University Press |year=1999 |isbn=978-0-521-65302-2 |url-access=registration}}
</ref><ref>
{{cite journal |last1=Jerrum |first1=M. |year=1986 |title=A compact representation of permutation groups |journal=J. Algorithms |volume=7 |pages=60–78 |doi=10.1016/0196-6774(86)90038-6 |s2cid=18896625 |number=1}}
</ref>
In this notation, the permutation is often written as an exponent, so ''σ'' acting on ''x'' is written ''x''<sup>''σ''</sup>; then the product is defined by <math>x^{\sigma\cdot\tau} = (x^\sigma)^\tau</math>. This article uses the first definition, where the rightmost permutation is applied first.

The [[function composition]] operation satisfies the axioms of a [[Group (mathematics)|group]]. It is [[Associative property|associative]], meaning <math>(\rho\sigma)\tau = \rho(\sigma\tau)</math>, and products of more than two permutations are usually written without parentheses. The composition operation also has an [[identity element]] (the identity permutation <math>\text{id}</math>), and each permutation <math>\sigma</math> has an inverse <math>\sigma^{-1}</math> (its [[inverse function]]) with <math>\sigma^{-1}\sigma = \sigma\sigma^{-1}=\text{id}</math>.