Generators - Maple Help

GroupTheory

 Generators
 obtain the generators of a group
 NonRedundantGenerators
 obtain the generators of a group, with redundant ones filtered out
 Labels
 obtain labels for the generators of a group

 Calling Sequence Generators(g) NonRedundantGenerators(g) Labels(g)

Parameters

 g - group data structure

Description

 • The Generators command returns a list of generators of a group.
 • The NonRedundantGenerators command returns a list of generators of a group, with redundant generators removed (no proper subset of the generators will generate the full group). This is typically a bit more time-consuming to compute than Generators.
 • The Labels command returns a list of labels for the generators of a group, or $\mathrm{undefined}$ if no labels are defined. In the first case, the labels are in the same order as the list that Generators returns.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{g1}≔\mathrm{Group}\left(\left\{a=\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right),b=\mathrm{Perm}\left(\left[\left[2,3\right]\right]\right),\mathrm{Perm}\left(\left[\left[3,4\right]\right]\right)\right\}\right)$
 ${\mathrm{g1}}{≔}⟨\left({1}{,}{2}\right){,}\left({2}{,}{3}\right){,}\left({3}{,}{4}\right)⟩$ (1)
 > $\mathrm{Generators}\left(\mathrm{g1}\right)$
 $\left[\left({1}{,}{2}\right){,}\left({2}{,}{3}\right){,}\left({3}{,}{4}\right)\right]$ (2)

Maple selects a label for the generator missing one.

 > $\mathrm{Labels}\left(\mathrm{g1}\right)$
 $\left[{a}{,}{b}{,}{\mathrm{_G}}\right]$ (3)

In this case, none of the generators are labeled, so Maple does not select any labels.

 > $\mathrm{g2}≔\mathrm{Group}\left(\left\{\left[\left[1,2\right]\right],\left[\left[2,3\right]\right],\left[\left[3,4\right]\right]\right\}\right)$
 ${\mathrm{g2}}{≔}⟨\left({1}{,}{2}\right){,}\left({2}{,}{3}\right){,}\left({3}{,}{4}\right)⟩$ (4)
 > $\mathrm{Labels}\left(\mathrm{g2}\right)$
 ${\mathrm{undefined}}$ (5)

Finitely presented groups use the names of the generators as labels.

 > $\mathrm{g3}≔\mathrm{Group}\left(\left\{a,b\right\},\left\{\left[a,a\right],\left[b,b\right],\left[a,b,a,b\right]\right\}\right)$
 ${\mathrm{g3}}{≔}⟨{}{a}{,}{b}{}{\mid }{}{{a}}^{{2}}{,}{{b}}^{{2}}{,}{a}{}{b}{}{a}{}{b}{}⟩$ (6)
 > $\mathrm{Labels}\left(\mathrm{g3}\right)$
 $\left[{a}{,}{b}\right]$ (7)

A Cayley table groups returns all its elements as its list of generators.

 > $\mathrm{g4}≔\mathrm{Group}\left(⟨⟨1|2|3⟩,⟨2|3|1⟩,⟨3|1|2⟩⟩\right)$
 ${\mathrm{g4}}{≔}{\mathrm{< a Cayley table group with 3 elements >}}$ (8)
 > $\mathrm{Generators}\left(\mathrm{g4}\right)$
 $\left[{1}{,}{2}{,}{3}\right]$ (9)
 > $\mathrm{NonRedundantGenerators}\left(\mathrm{g4}\right)$
 $\left[{2}\right]$ (10)

Compatibility

 • The GroupTheory[Generators], GroupTheory[NonRedundantGenerators] and GroupTheory[Labels] commands were introduced in Maple 17.