The procedure Group creates data structures representing permutation groups, groups given by generators and relations, and Cayley table groups. The procedures PermutationGroup, FPGroup, CustomGroup, and CayleyTableGroup are specialized procedures that create only groups of the type indicated by their name. Group accepts all calling sequences for the other constructors.

The first calling sequence above creates a permutation group. See PermutationGroup for more details on this calling sequence.

The second calling sequence creates a finitely presented group. See FPGroup for more details on this calling sequence.

The third calling sequence creates a group given by custom operations. See CustomGroup for more details on this calling sequence.

The fourth calling sequence creates a group given by a Cayley table. See CayleyTableGroup for more details on this calling sequence.

The final calling sequence creates either a permutation group or a finitely presented group from a group created by permgroup or grelgroup or subgrel. This is available as a backwards compatibility option.

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

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

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

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

$12$

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

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

$\mathrm{g3}≔\mathrm{Group}\left(\left\{a\,b\right\}\,\left[\left[a\,a\,a\right]\,\left[b\,b\,b\right]\,\left[a\,b\,\frac{1}{a}\,\frac{1}{b}\right]\right]\right)$

$\mathrm{g3}≔⟨a\,b\mid a^3\,b^3\,ab{a}^{-1}{b}^{-1}⟩$

$\mathrm{g4}≔\mathrm{Group}\left(\left[c\right]\,\left\{\left[c\,c\,c\right]\right\}\,'\mathrm{supergroup}'=\mathrm{g3}\,'\mathrm{embedding}'=\left[\left[a\,b\,a\right]\right]\right)$

$\mathrm{g4}≔⟨c\mid c^3⟩$

$\mathrm{g5}≔\mathrm{Subgroup}\left(\left[\left[a\,b\,a\right]\right]\,\mathrm{g3}\right)$

$\mathrm{g5}≔⟨\mathrm{\_G}\mid {\mathrm{\_G}}^3⟩$

$\mathrm{operation\_module}≔\mathbf{module}\left(\right)\mathrm{\_export}\left(\mathrm{`.`}≔\left(a\,b\right)\→a\+b\mathbf{mod}6\right)\;\mathrm{\_export}\left(\mathrm{`/`}≔a\→\−a\mathbf{mod}6\right)\mathbf{end\; module}$

$\mathrm{operation\_module}≔\mathbf{module}\left(\right)\mathbf{export}\mathrm{`.`}\,\mathrm{`/`}\;\mathbf{end\; module}$

$\mathrm{g6}≔\mathrm{Group}\left(\left[1\right]\,\mathrm{operation\_module}\right)$

$\mathrm{g6}≔\mathrm{<\; a\; custom\; group\; with\; 1\; generator\; >}$

$\mathrm{equal\_modulo\_6}≔\left(a\&comma;b\right)\mapsto \mathrm{irem}\left(b-a\&comma;6\right)=0$

$\mathrm{equal\_modulo\_6}≔\left(a\&comma;b\right)\mapsto \mathrm{irem}\left(b-a\&comma;6\right)=0$

$\mathrm{less\_modulo\_6}≔\left(a\&comma;b\right)\mapsto a\mathbf{mod}6<b\mathbf{mod}6$

$\mathrm{less\_modulo\_6}≔\left(a\&comma;b\right)\mapsto a\mathbf{mod}6<b\mathbf{mod}6$

$\mathrm{g7}≔\mathrm{Group}\left(\left[1\right]\&comma;\mathrm{`.`}=\mathrm{`+`}\&comma;\mathrm{`/`}=\mathrm{`-`}\&comma;\mathrm{`=`}=\mathrm{equal\_modulo\_6}\&comma;\mathrm{`<`}=\mathrm{less\_modulo\_6}\right)$

$\mathrm{g7}≔\mathrm{<\; a\; custom\; group\; with\; 1\; generator\; >}$

$\mathrm{g8}≔\mathrm{Group}\left(⟨⟨1\left|2\right|3|4⟩\&comma;⟨2\left|1\right|4|3⟩\&comma;⟨3\left|4\right|1|2⟩\&comma;⟨4\left|3\right|2|1⟩⟩\&comma;\mathrm{labels}=\left[a\&comma;b\&comma;c\&comma;d\right]\right)$

$\mathrm{g8}≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 4\; elements\; >}$

$\mathrm{g9}≔\mathrm{Group}\left(\mathrm{permgroup}\left(5\&comma;\left\{\left[\left[1\&comma;2\&comma;3\&comma;4\right]\right]\right\}\right)\right)$

$\mathrm{g9}≔⟨\left(1\&comma;2\&comma;3\&comma;4\right)⟩$

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

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