The FPGroup command creates a finitely presented group data structure. Finitely presented groups are often difficult to work with computationally, and sometimes it may be easier to create a permutation representation of your group using the converting calling sequence of the PermutationGroup command if the group you are working with is finite.

The first calling sequence creates a finitely presented group using the following arguments. generators is a list or set of names of generators for the group. relators is a list or set of relators that is imposed on these generators (i.e., the set of relations that is factored out). Each relator is given as a list, each element of which is either a member of generators or an inverse of these generators.

To specify a subgroup of a finitely presented group, you can include an $\mathrm{sgopt}$ argument - an equation of the form $\mathrm{supergroup}=g$ - and an $\mathrm{embopt}$ argument - an equation of the form $\mathrm{embedding}=t$. The supergroup itself is given by $g$, and $t$ specifies how this subgroup embeds into its supergroup: it is a list of words expressing the value of each of the generators in the generators of $g$ and their inverses. It may be easier to construct a subgroup with the Subgroup command, which uses this calling sequence itself.

The second calling sequence is a syntax shortcut available when the GroupTheory package has been loaded (using with). This calling sequence will work if there is at least one generator, and at least one of the relators is expressed as an equation.

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

$\mathrm{g1}≔\mathrm{FPGroup}\left(\left\{a\,b\right\}\,\left[\left[a\,a\,a\right]\,\left[b\,b\,b\right]\,\left[a\,b\,\frac{1}{a}\,\frac{1}{b}\right]\right]\right)$

$\mathrm{g1}≔⟨a\,b\mid a^3\,b^3\,ab{a}^{-1}{b}^{-1}⟩$

$\mathrm{g2}≔\mathrm{FPGroup}\left(\left[c\right]\,\left\{\left[c\,c\,c\right]\right\}\,'\mathrm{supergroup}'=\mathrm{g1}\,'\mathrm{embedding}'=\left[\left[a\,b\,a\right]\right]\right)$

$\mathrm{g2}≔⟨c\mid c^3⟩$

$\mathrm{g3}≔\mathrm{Subgroup}\left(\left[\left[a\,b\,a\right]\right]\,\mathrm{g1}\right)$

$\mathrm{g3}≔⟨\mathrm{\_G}\mid {\mathrm{\_G}}^3⟩$

$\mathrm{g4}≔⟨⟨a\,b⟩|⟨a^2=1\,b^3=1\,a·b·a=b^2⟩⟩$

$\mathrm{g4}≔⟨a\,b\mid a^2\,b^3\,a^{-1}{b}^{-1}{a}^{-1}{b}^2⟩$

$\mathrm{GroupOrder}\left(\mathrm{g4}\right)$

$6$

$\mathrm{g5}≔⟨⟨a\,b⟩|⟨a^8\,b^2\,{\left(a^4·b\right)}^8\,\left[b\,b^a\right]=1\,\left[b\,b^{a^2}\right]\,\left[b\,b^{a^3}\right]=1⟩⟩$

$\mathrm{g5}≔⟨a\,b\mid b^2\,a^8\,b^{-1}{a}^{-1}{b}^{-1}ab{a}^{-1}ba\,b^{-1}{a}^{-2}{b}^{-1}{a}^{2}b{a}^{-2}b{a}^2\,b^{-1}{a}^{-3}{b}^{-1}{a}^{3}b{a}^{-3}b{a}^3\,a^{4}b{a}^{4}b{a}^{4}b{a}^{4}b{a}^{4}b{a}^{4}b{a}^{4}b{a}^{4}b⟩$

$\mathrm{GroupOrder}\left(\mathrm{g5}\right)$

$32768$

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

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