GroupTheory/NormalSubgroups - Maple Help

GroupTheory

 NormalSubgroups
 compute the normal subgroups of a finite group

 Calling Sequence NormalSubgroups( G )

Parameters

 G - a finite group

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $S≔\mathrm{NormalSubgroups}\left(G\right)$
 ${S}{≔}\left[⟨\left({2}{,}{3}{,}{4}\right){,}\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right)⟩{,}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right)⟩{,}⟨⟩\right]$ (2)
 > $\mathrm{andmap}\left(\mathrm{IsNormal},S,G\right)$
 ${\mathrm{true}}$ (3)

The alternating group of degree 5 is simple, so it has only two normal subgroups, itself and the trivial subgroup.

 > $G≔\mathrm{Alt}\left(5\right)$
 ${G}{≔}{{\mathbf{A}}}_{{5}}$ (4)
 > $\mathrm{NormalSubgroups}\left(G\right)$
 $\left[{{\mathbf{A}}}_{{5}}{,}⟨⟩\right]$ (5)

The automorphism group of the Clebsch graph contains a perfect normal subgroup of index two.

 > $\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{GraphTheory}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A≔\mathrm{AutomorphismGroup}\left(\mathrm{SpecialGraphs}:-\mathrm{ClebschGraph}\left(\right)\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{GroupOrder}\left(A\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{NA}≔\mathrm{NormalSubgroups}\left(A\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{GroupOrder}\left({\mathrm{NA}}_{2}\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{IsPerfect}\left({\mathrm{NA}}_{2}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}$
 ${\mathrm{true}}$ (6)

Compatibility

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