The NormalSubgroups( G ) command computes the normal subgroups of a finite group G.

The group G must be an instance of a permutation group or a Cayley table group.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$S≔\mathrm{NormalSubgroups}\left(G\right)$

$S≔\left[⟨\left(2\,3\,4\right)\,\left(1\,2\right)\left(3\,4\right)\,\left(1\,3\right)\left(2\,4\right)⟩\,⟨\left(1\,2\right)\left(3\,4\right)\,\left(1\,3\right)\left(2\,4\right)⟩\,⟨⟩\right]$

$\mathrm{andmap}\left(\mathrm{IsNormal}\,S\,G\right)$

$\mathrm{true}$

$G≔\mathrm{Alt}\left(5\right)$

$G≔{\mathbf{A}}_5$

$\mathrm{NormalSubgroups}\left(G\right)$

$\left[{\mathbf{A}}_5\,⟨⟩\right]$

$\mathbf{use}\mathrm{GraphTheory}\mathbf{in}A≔\mathrm{AutomorphismGroup}\left(\mathrm{SpecialGraphs}:-\mathrm{ClebschGraph}\left(\right)\right)\;\mathrm{GroupOrder}\left(A\right)\;\mathrm{NA}≔\mathrm{NormalSubgroups}\left(A\right)\;\mathrm{GroupOrder}\left({\mathrm{NA}}_2\right)\;\mathrm{IsPerfect}\left({\mathrm{NA}}_2\right)\mathbf{end\; use}$

$\mathrm{true}$

The GroupTheory[NormalSubgroups] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.