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.

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

$\mathrm{g1}≔\mathrm{Group}\left(\left\{a=\mathrm{Perm}\left(\left[\left[1\,2\right]\right]\right)\,b=\mathrm{Perm}\left(\left[\left[2\,3\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[3\,4\right]\right]\right)\right\}\right)$

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

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

$\left[\left(1\,2\right)\,\left(2\,3\right)\,\left(3\,4\right)\right]$

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

$\left[a\,b\,\mathrm{\_G}\right]$

$\mathrm{g2}≔\mathrm{Group}\left(\left\{\left[\left[1\,2\right]\right]\,\left[\left[2\,3\right]\right]\,\left[\left[3\,4\right]\right]\right\}\right)$

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

$\mathrm{Labels}\left(\mathrm{g2}\right)$

$\mathrm{undefined}$

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

$\mathrm{g3}≔⟨a\,b\mid a^2\,b^2\,abab⟩$

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

$\left[a\,b\right]$

$\mathrm{g4}≔\mathrm{Group}\left(⟨⟨1\left|2\right|3⟩\,⟨2\left|3\right|1⟩\,⟨3\left|1\right|2⟩⟩\right)$

$\mathrm{g4}≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 3\; elements\; >}$

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

$\left[1\&comma;2\&comma;3\right]$

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

$\left[2\right]$

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

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