The PermutationGroup command creates a permutation group data structure. Permutation groups are implemented relatively efficiently, and if there is a natural permutation representation of a group you might want to compute with, this is the implementation of choice.

The first calling sequence above uses the following arguments. The first argument consists of permutations generating the group. Each permutation is given as a list of disjoint cycles; each cycle is represented by a list of its elements. If the degree is not given, it defaults to the largest point moved by any of the generating permutations.

Optionally, generating permutations can be given by equations of the form $\mathrm{name}=\mathrm{permutation}$. The permutations themselves are given in disjoint cycle notation, as before. This associates a name to each permutation, which can be retrieved using the Labels command. If any generators are named, Maple automatically generates names for the generators for which no name is given.

You can optionally use an $\mathrm{sgopt}$ argument - an equation of the form $\mathrm{supergroup}=g$ - to indicate a supergroup that the given group is a subgroup of. The given generators should be members of $g$. Maple performs a superficial (but not exhaustive) test to check if this is the case if a supergroup is specified. If no supergroup is specified or the user specifies $\mathrm{supergroup}=\mathrm{undefined}$, then Maple does not mark the resulting group as a subgroup of any particular group.

The second calling sequence takes a group data structure constructed by the GroupTheory package and converts it to a permutation group. The names given to the generators of the result, retrievable using the Labels command, are their preimages in $\mathrm{gp}$.

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

$\mathrm{g1}≔\mathrm{PermutationGroup}\left(\left\{\left[\left[1\,2\right]\right]\,\left[\left[1\,2\,3\right]\,\left[4\,5\right]\right]\right\}\right)$

$\mathrm{g1}≔⟨\left(1\,2\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩$

$\mathrm{GroupOrder}\left(\mathrm{g1}\right)$

$12$

$\mathrm{g2}≔\mathrm{PermutationGroup}\left(a=\left[\left[1\,2\right]\right]\,b=\left[\left[1\,2\,3\right]\,\left[4\,5\right]\right]\,'\mathrm{degree}'=6\,'\mathrm{supergroup}'=\mathrm{PermutationGroup}\left(\left\{\left[\left[1\,2\right]\right]\,\left[\left[1\,2\,3\,4\,5\,6\right]\right]\right\}\right)\right)$

$\mathrm{g2}≔⟨\left(1\,2\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩$

$\mathrm{g3}≔\mathrm{Group}\left(⟨⟨1\left|2\right|3|4⟩\,⟨2\left|1\right|4|3⟩\,⟨3\left|4\right|1|2⟩\,⟨4\left|3\right|2|1⟩⟩\,\mathrm{labels}=\left[a\,b\,c\,d\right]\right)$

$\mathrm{g3}≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 4\; elements\; >}$

$\mathrm{PermutationGroup}\left(\mathrm{g3}\right)$

$⟨\left(1\&comma;2\right)\left(3\&comma;4\right)\&comma;\left(1\&comma;3\right)\left(2\&comma;4\right)\&comma;\left(1\&comma;4\right)\left(2\&comma;3\right)⟩$

The GroupTheory[PermutationGroup] command was introduced in Maple 17.

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