A permutation is a bijective mapping from the set$\left\{1\,2\,\dots \,n\right\}$ to itself, for some positive integer $n$.

The set of all such permutations forms the symmetric group of degree $n$, and subgroups of symmetric groups are permutation groups.

Permutations are typically represented as products of disjoint cycles, each of which is an orbit of the permutation.  This is a list of the form$\left[c_1\,c_2\,\dots \,c_k\right]$ in which each $c_i$ is itself a list$\left[i_1\,i_2\,\dots \,i_m\right]$ representing a cycle of the form$i_1\mapsto i_2\mapsto i_m\mapsto i_1$ .

The Perm constructor creates a permutation, given a specification of its disjoint cycle structure in the form of a list of lists.  You can also use a permutation list, which is just the representation of the permutation as a list L of points in which L[ i ] specifies the image of i under the permutation. In particular, the identity permutation is represented by the expression Perm([]).

The Permutation Operations in GroupTheory page lists commands that operate on permutation objects and are part of the GroupTheory package.

Note that the non-commutative multiplication operator . can be used to multiply permutations.

$a≔\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\,5\right]\right]\right)$

$a≔\left(1\,2\right)\left(3\,4\,5\right)$

$a\left[1\right]$

$2$

$a\left[2\right]$

$1$

$a\left[3\right]$

$4$

$a\left[4\right]$

$5$

$a\left[5\right]$

$3$

$b≔\mathrm{Perm}\left(\left[\left[1\,3\right]\,\left[2\,6\right]\right]\right)$

$b≔\left(1\,3\right)\left(2\,6\right)$

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{PermDegree}\left(a\right)$

$5$

$\mathrm{PermDegree}\left(b\right)$

$6$

$\mathrm{PermProduct}\left(a\,b\right)$

$\left(1\,6\,2\,3\,4\,5\right)$

The Perm command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.