Editing Permutation group (section)

==Neutral element and inverses==

The identity permutation, which maps every element of the set to itself, is the [[neutral element]] for this product. In two-line notation, the identity is
:<math>\begin{pmatrix}1 & 2 & 3 & \cdots & n \\ 1 & 2 & 3 & \cdots & n\end{pmatrix}.</math>
In cycle notation, ''e'' = (1)(2)(3)...(''n'') which by convention is also denoted by just (1) or even ().<ref>{{harvnb|Rotman|2006|loc=p. 108}}</ref>

Since [[bijections]] have [[Inverse function|inverses]], so do permutations, and the inverse ''σ''<sup>−1</sup> of ''σ'' is again a permutation. Explicitly, whenever ''σ''(''x'')=''y'' one also has ''σ''<sup>−1</sup>(''y'')=''x''. In two-line notation the inverse can be obtained by interchanging the two lines (and sorting the columns if one wishes the first line to be in a given order). For instance
:<math>\begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end{pmatrix}^{-1}
      =\begin{pmatrix}2 & 5 & 4 & 3 & 1\\ 1 & 2 & 3 & 4 & 5 \end{pmatrix}
      =\begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 5 & 1 & 4 & 3 & 2\end{pmatrix}.</math>

To obtain the inverse of a single cycle, we reverse the order of its elements. Thus,
:<math> (1 2 5)^{-1} = (5 2 1) = (152).</math>

To obtain the inverse of a product of cycles, we first reverse the order of the cycles, and then we take the inverse of each as above. Thus,
:<math> [(1 2 5)(3 4)]^{-1} = (34)^{-1}(125)^{-1} = (43)(521) = (34)(152).</math>

Having an associative product, an identity element, and inverses for all its elements, makes the set of all permutations of ''M'' into a [[group (mathematics)|group]], Sym(''M''); a permutation group.