 Group - Maple Help

GroupTheory

 Group
 create a group data structure Calling Sequence Group(permutations, degreeopt, sgopt) Group(generators, relators, embopt, sgopt) Group(gens, operations, customopts, sgopt, lbopt) Group(m, sgopt, embopt, lbopt) Group(g) Parameters

 permutations - set, list, or sequence of permutations represented as disjoint cycles, optionally with names degreeopt - (optional) equation of the form $\mathrm{degree}=n$ where $n$ is a nonnegative integer; number of letters on which the permutation group acts sgopt - (optional) equation of the form $\mathrm{supergroup}=g$ where $g$ is a supergroup of the group to be constructed; constructs the given group as a subgroup of $g$ generators - set or list of names relators - set or list of relators, each represented as a list of the $\mathrm{generators}$ names and their inverses gens - set or list of custom group generators operations - (optional) module describing group operations customopts - (optional) equations of the form one = e, 1 = e, multiply = m, . = m, inverse = i, / = i, equals = eq, = = eq, less = o, < = o, or print = p lbopt - (optional) equation of the form $\mathrm{labels}=\mathrm{lst}$, giving labels for the generators m - Cayley table defining group operations embopt - (optional) equation of the form embedding = t, where t gives the embedding g - group given by permgroup or grelgroup or subgrel Description

 • 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. Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{g1}≔\mathrm{Group}\left(\left\{\left[\left[1,2\right]\right],\left[\left[1,2,3\right],\left[4,5\right]\right]\right\}\right)$
 ${\mathrm{g1}}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩$ (1)
 > $\mathrm{GroupOrder}\left(\mathrm{g1}\right)$
 ${12}$ (2)
 > $\mathrm{g2}≔\mathrm{Group}\left(a=\left[\left[1,2\right]\right],b=\left[\left[1,2,3\right],\left[4,5\right]\right],'\mathrm{degree}'=6,'\mathrm{supergroup}'=\mathrm{Group}\left(\left\{\left[\left[1,2\right]\right],\left[\left[1,2,3,4,5,6\right]\right]\right\}\right)\right)$
 ${\mathrm{g2}}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩$ (3)

A finitely presented group and a subgroup of it, in two ways:

 > $\mathrm{g3}≔\mathrm{Group}\left(\left\{a,b\right\},\left[\left[a,a,a\right],\left[b,b,b\right],\left[a,b,\frac{1}{a},\frac{1}{b}\right]\right]\right)$
 ${\mathrm{g3}}{≔}⟨{}{a}{,}{b}{}{\mid }{}{{a}}^{{3}}{,}{{b}}^{{3}}{,}{a}{}{b}{}{{a}}^{{-1}}{}{{b}}^{{-1}}{}⟩$ (4)
 > $\mathrm{g4}≔\mathrm{Group}\left(\left[c\right],\left\{\left[c,c,c\right]\right\},'\mathrm{supergroup}'=\mathrm{g3},'\mathrm{embedding}'=\left[\left[a,b,a\right]\right]\right)$
 ${\mathrm{g4}}{≔}⟨{}{c}{}{\mid }{}{{c}}^{{3}}{}⟩$ (5)
 > $\mathrm{g5}≔\mathrm{Subgroup}\left(\left[\left[a,b,a\right]\right],\mathrm{g3}\right)$
 ${\mathrm{g5}}{≔}⟨{}{\mathrm{_G}}{}{\mid }{}{{\mathrm{_G}}}^{{3}}{}⟩$ (6)

The additive group of the integers modulo $6$ can be constructed as a custom group in two ways: reducing to the range $0..5$ after every operation or not. We present both ways here. We also show the two ways of specifying the operations: with equations and with a module.

 > $\mathrm{operation_module}≔\mathbf{module}\left(\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{_export}\left(\mathrm{.}≔\left(a,b\right)→a+b\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}6\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{_export}\left(\mathrm{/}≔a→−a\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}6\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end module}$
 ${\mathrm{operation_module}}{≔}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{export}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{.}}{,}{\mathrm{/}}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (7)
 > $\mathrm{g6}≔\mathrm{Group}\left(\left[1\right],\mathrm{operation_module}\right)$
 ${\mathrm{g6}}{≔}{\mathrm{< a custom group with 1 generator >}}$ (8)
 > $\mathrm{equal_modulo_6}≔\left(a,b\right)↦\mathrm{irem}\left(b-a,6\right)=0$
 ${\mathrm{equal_modulo_6}}{≔}\left({a}{,}{b}\right){↦}{\mathrm{irem}}{}\left({b}{-}{a}{,}{6}\right){=}{0}$ (9)
 > $\mathrm{less_modulo_6}≔\left(a,b\right)↦a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}6
 ${\mathrm{less_modulo_6}}{≔}\left({a}{,}{b}\right){↦}{a}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{mod}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{6}{<}{b}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{mod}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{6}$ (10)
 > $\mathrm{g7}≔\mathrm{Group}\left(\left[1\right],\mathrm{.}=\mathrm{+},\mathrm{/}=\mathrm{-},\mathrm{=}=\mathrm{equal_modulo_6},\mathrm{<}=\mathrm{less_modulo_6}\right)$
 ${\mathrm{g7}}{≔}{\mathrm{< a custom group with 1 generator >}}$ (11)
 > $\mathrm{g8}≔\mathrm{Group}\left(⟨⟨1|2|3|4⟩,⟨2|1|4|3⟩,⟨3|4|1|2⟩,⟨4|3|2|1⟩⟩,\mathrm{labels}=\left[a,b,c,d\right]\right)$
 ${\mathrm{g8}}{≔}{\mathrm{< a Cayley table group with 4 elements >}}$ (12)
 > $\mathrm{g9}≔\mathrm{Group}\left(\mathrm{permgroup}\left(5,\left\{\left[\left[1,2,3,4\right]\right]\right\}\right)\right)$
 ${\mathrm{g9}}{≔}⟨\left({1}{,}{2}{,}{3}{,}{4}\right)⟩$ (13) Compatibility

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