MaximalNormalSubgroups - Maple Help
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GroupTheory

 NormalSubgroups
 compute the normal subgroups of a finite group
 MinimalNormalSubgroups
 compute the minimal normal subgroups of a finite group
 MaximalNormalSubgroups
 compute the maximal normal subgroups of a finite group

 Calling Sequence NormalSubgroups( G ) MinimalNormalSubgroups( G ) MaximalNormalSubgroups( 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.
 • The MinimalNormalSubgroups( G ) command computes the minimal normal subgroups of a permutation group G. These are the non-trivial normal subgroups of G that properly contain no other non-trivial normal subgroup of G.
 • The MaximalNormalSubgroups( G ) command computes the maximal normal subgroups of a permutation group G. These subgroups are proper normal subgroups of G contained properly in no other proper normal subgroup of G.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $S≔\mathrm{NormalSubgroups}\left(G\right)$
 ${S}{≔}\left[⟨\left({1}{,}{4}\right)\left({2}{,}{3}\right){,}\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({2}{,}{3}{,}{4}\right)⟩{,}⟨\left({1}{,}{4}\right)\left({2}{,}{3}\right){,}\left({1}{,}{2}\right)\left({3}{,}{4}\right)⟩{,}⟨⟩\right]$ (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)
 > $G≔\mathrm{DihedralGroup}\left(10\right)$
 ${G}{≔}{{\mathbf{D}}}_{{10}}$ (6)
 > $L≔\mathrm{NormalSubgroups}\left(G\right)$
 ${L}{≔}\left[⟨\left({1}{,}{6}\right)\left({2}{,}{5}\right)\left({3}{,}{4}\right)\left({7}{,}{10}\right)\left({8}{,}{9}\right){,}\left({1}{,}{3}\right)\left({4}{,}{10}\right)\left({5}{,}{9}\right)\left({6}{,}{8}\right){,}\left({1}{,}{8}\right)\left({2}{,}{7}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right)\left({9}{,}{10}\right)⟩{,}⟨\left({1}{,}{6}\right)\left({2}{,}{5}\right)\left({3}{,}{4}\right)\left({7}{,}{10}\right)\left({8}{,}{9}\right){,}\left({1}{,}{8}\right)\left({2}{,}{7}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right)\left({9}{,}{10}\right)⟩{,}⟨\left({1}{,}{9}{,}{7}{,}{5}{,}{3}\right)\left({2}{,}{10}{,}{8}{,}{6}{,}{4}\right){,}\left({1}{,}{3}\right)\left({4}{,}{10}\right)\left({5}{,}{9}\right)\left({6}{,}{8}\right)⟩{,}⟨\left({1}{,}{9}{,}{7}{,}{5}{,}{3}\right)\left({2}{,}{10}{,}{8}{,}{6}{,}{4}\right){,}\left({1}{,}{10}{,}{9}{,}{8}{,}{7}{,}{6}{,}{5}{,}{4}{,}{3}{,}{2}\right)⟩{,}⟨\left({1}{,}{9}{,}{7}{,}{5}{,}{3}\right)\left({2}{,}{10}{,}{8}{,}{6}{,}{4}\right)⟩{,}⟨\left({1}{,}{6}\right)\left({2}{,}{7}\right)\left({3}{,}{8}\right)\left({4}{,}{9}\right)\left({5}{,}{10}\right)⟩{,}⟨⟩\right]$ (7)
 > $\mathrm{map}\left(\mathrm{GroupOrder},L\right)$
 $\left[{20}{,}{10}{,}{10}{,}{10}{,}{5}{,}{2}{,}{1}\right]$ (8)
 > $\mathrm{map}\left(\mathrm{GroupOrder},\mathrm{MinimalNormalSubgroups}\left(G\right)\right)$
 $\left[{2}{,}{5}\right]$ (9)
 > $\mathrm{map}\left(\mathrm{GroupOrder},\mathrm{MaximalNormalSubgroups}\left(G\right)\right)$
 $\left[{10}{,}{10}{,}{10}\right]$ (10)

Observe that the trivial group has neither maximal nor minimal normal subgroups.

 > $\left(\mathrm{MinimalNormalSubgroups},\mathrm{MaximalNormalSubgroups}\right)\left(\mathrm{TrivialGroup}\left(\right)\right)$
 $\left[\right]{,}\left[\right]$ (11)

The only maximal normal subgroup of a simple group is the trivial subgroup.

 > $\mathrm{MaximalNormalSubgroups}\left(\mathrm{Suzuki2B2}\left(32\right)\right)$
 $\left[⟨⟩\right]$ (12)

Moreover, the only minimal normal subgroup of a simple group is the entire group itself.

 > $\mathrm{MinimalNormalSubgroups}\left(\mathrm{McLaughlinGroup}\left(\right)\right)$
 $\left[{McL}\right]$ (13)

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

 > $\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{GraphTheory}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A≔\mathrm{AutomorphismGroup}\left(\mathrm{SpecialGraphs}:-\mathrm{ClebschGraph}\left(\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}$
 $⟨\left({4}{,}{13}\right)\left({5}{,}{6}\right)\left({8}{,}{14}\right)\left({9}{,}{11}\right){,}\left({2}{,}{4}\right)\left({5}{,}{16}\right)\left({11}{,}{15}\right)\left({12}{,}{14}\right){,}\left({1}{,}{2}{,}{5}{,}{3}{,}{9}{,}{11}{,}{12}{,}{8}\right)\left({4}{,}{14}{,}{6}{,}{13}{,}{15}{,}{16}{,}{7}{,}{10}\right){,}\left({3}{,}{7}\right)\left({5}{,}{14}\right)\left({6}{,}{8}\right)\left({12}{,}{16}\right){,}\left({2}{,}{5}\right)\left({4}{,}{16}\right)\left({7}{,}{9}\right)\left({8}{,}{10}\right)⟩$ (14)
 > $\mathrm{GroupOrder}\left(A\right)$
 ${1920}$ (15)
 > $\mathrm{NA}≔\mathrm{NormalSubgroups}\left(A\right)$
 ${\mathrm{NA}}{≔}\left[⟨\left({1}{,}{3}{,}{8}{,}{13}{,}{11}\right)\left({2}{,}{7}{,}{12}{,}{16}{,}{6}\right)\left({5}{,}{15}{,}{14}{,}{9}{,}{10}\right){,}\left({1}{,}{3}{,}{12}{,}{6}{,}{13}{,}{11}\right)\left({2}{,}{7}{,}{8}\right)\left({5}{,}{9}{,}{10}\right)\left({14}{,}{15}\right){,}\left({1}{,}{3}{,}{14}{,}{4}{,}{9}\right)\left({2}{,}{7}{,}{12}{,}{16}{,}{5}\right)\left({6}{,}{15}{,}{8}{,}{11}{,}{10}\right)⟩{,}⟨\left({1}{,}{3}{,}{8}{,}{13}{,}{11}\right)\left({2}{,}{7}{,}{12}{,}{16}{,}{6}\right)\left({5}{,}{15}{,}{14}{,}{9}{,}{10}\right){,}\left({1}{,}{3}{,}{14}{,}{4}{,}{9}\right)\left({2}{,}{7}{,}{12}{,}{16}{,}{5}\right)\left({6}{,}{15}{,}{8}{,}{11}{,}{10}\right)⟩{,}⟨\left({1}{,}{5}\right)\left({2}{,}{6}\right)\left({3}{,}{4}\right)\left({7}{,}{11}\right)\left({8}{,}{15}\right)\left({9}{,}{12}\right)\left({10}{,}{14}\right)\left({13}{,}{16}\right){,}\left({1}{,}{6}\right)\left({2}{,}{5}\right)\left({3}{,}{13}\right)\left({4}{,}{16}\right)\left({7}{,}{9}\right)\left({8}{,}{10}\right)\left({11}{,}{12}\right)\left({14}{,}{15}\right){,}\left({1}{,}{14}\right)\left({2}{,}{8}\right)\left({3}{,}{11}\right)\left({4}{,}{7}\right)\left({5}{,}{10}\right)\left({6}{,}{15}\right)\left({9}{,}{16}\right)\left({12}{,}{13}\right){,}\left({1}{,}{16}\right)\left({2}{,}{3}\right)\left({4}{,}{6}\right)\left({5}{,}{13}\right)\left({7}{,}{15}\right)\left({8}{,}{11}\right)\left({9}{,}{14}\right)\left({10}{,}{12}\right)⟩{,}⟨⟩\right]$ (16)
 > $\mathrm{GroupOrder}\left(\mathrm{NA}\left[2\right]\right)$
 ${960}$ (17)
 > $\mathrm{IsPerfect}\left(\mathrm{NA}\left[2\right]\right)$
 ${\mathrm{true}}$ (18)
 > $\mathrm{IsPrimitive}\left(\mathrm{NA}\left[2\right]\right)$
 ${\mathrm{true}}$ (19)

Since every subgroup of an abelian group is normal, the following example returns the collection of all subgroups of the group.

 > $L≔\mathrm{NormalSubgroups}\left(\mathrm{CyclicGroup}\left(36\right)\right)$
 ${L}{≔}\left[⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}{,}{9}{,}{10}{,}{11}{,}{12}{,}{13}{,}{14}{,}{15}{,}{16}{,}{17}{,}{18}{,}{19}{,}{20}{,}{21}{,}{22}{,}{23}{,}{24}{,}{25}{,}{26}{,}{27}{,}{28}{,}{29}{,}{30}{,}{31}{,}{32}{,}{33}{,}{34}{,}{35}{,}{36}\right)⟩{,}⟨\left({1}{,}{3}{,}{5}{,}{7}{,}{9}{,}{11}{,}{13}{,}{15}{,}{17}{,}{19}{,}{21}{,}{23}{,}{25}{,}{27}{,}{29}{,}{31}{,}{33}{,}{35}\right)\left({2}{,}{4}{,}{6}{,}{8}{,}{10}{,}{12}{,}{14}{,}{16}{,}{18}{,}{20}{,}{22}{,}{24}{,}{26}{,}{28}{,}{30}{,}{32}{,}{34}{,}{36}\right)⟩{,}⟨\left({1}{,}{4}{,}{7}{,}{10}{,}{13}{,}{16}{,}{19}{,}{22}{,}{25}{,}{28}{,}{31}{,}{34}\right)\left({2}{,}{5}{,}{8}{,}{11}{,}{14}{,}{17}{,}{20}{,}{23}{,}{26}{,}{29}{,}{32}{,}{35}\right)\left({3}{,}{6}{,}{9}{,}{12}{,}{15}{,}{18}{,}{21}{,}{24}{,}{27}{,}{30}{,}{33}{,}{36}\right)⟩{,}⟨\left({1}{,}{5}{,}{9}{,}{13}{,}{17}{,}{21}{,}{25}{,}{29}{,}{33}\right)\left({2}{,}{6}{,}{10}{,}{14}{,}{18}{,}{22}{,}{26}{,}{30}{,}{34}\right)\left({3}{,}{7}{,}{11}{,}{15}{,}{19}{,}{23}{,}{27}{,}{31}{,}{35}\right)\left({4}{,}{8}{,}{12}{,}{16}{,}{20}{,}{24}{,}{28}{,}{32}{,}{36}\right)⟩{,}⟨\left({1}{,}{7}{,}{13}{,}{19}{,}{25}{,}{31}\right)\left({2}{,}{8}{,}{14}{,}{20}{,}{26}{,}{32}\right)\left({3}{,}{9}{,}{15}{,}{21}{,}{27}{,}{33}\right)\left({4}{,}{10}{,}{16}{,}{22}{,}{28}{,}{34}\right)\left({5}{,}{11}{,}{17}{,}{23}{,}{29}{,}{35}\right)\left({6}{,}{12}{,}{18}{,}{24}{,}{30}{,}{36}\right)⟩{,}⟨\left({1}{,}{10}{,}{19}{,}{28}\right)\left({2}{,}{11}{,}{20}{,}{29}\right)\left({3}{,}{12}{,}{21}{,}{30}\right)\left({4}{,}{13}{,}{22}{,}{31}\right)\left({5}{,}{14}{,}{23}{,}{32}\right)\left({6}{,}{15}{,}{24}{,}{33}\right)\left({7}{,}{16}{,}{25}{,}{34}\right)\left({8}{,}{17}{,}{26}{,}{35}\right)\left({9}{,}{18}{,}{27}{,}{36}\right)⟩{,}⟨\left({1}{,}{13}{,}{25}\right)\left({2}{,}{14}{,}{26}\right)\left({3}{,}{15}{,}{27}\right)\left({4}{,}{16}{,}{28}\right)\left({5}{,}{17}{,}{29}\right)\left({6}{,}{18}{,}{30}\right)\left({7}{,}{19}{,}{31}\right)\left({8}{,}{20}{,}{32}\right)\left({9}{,}{21}{,}{33}\right)\left({10}{,}{22}{,}{34}\right)\left({11}{,}{23}{,}{35}\right)\left({12}{,}{24}{,}{36}\right)⟩{,}⟨\left({1}{,}{19}\right)\left({2}{,}{20}\right)\left({3}{,}{21}\right)\left({4}{,}{22}\right)\left({5}{,}{23}\right)\left({6}{,}{24}\right)\left({7}{,}{25}\right)\left({8}{,}{26}\right)\left({9}{,}{27}\right)\left({10}{,}{28}\right)\left({11}{,}{29}\right)\left({12}{,}{30}\right)\left({13}{,}{31}\right)\left({14}{,}{32}\right)\left({15}{,}{33}\right)\left({16}{,}{34}\right)\left({17}{,}{35}\right)\left({18}{,}{36}\right)⟩{,}⟨⟩\right]$ (20)
 > $L≔\mathrm{MaximalNormalSubgroups}\left(\mathrm{CyclicGroup}\left(100000\right)\right):$
 > $\mathrm{map}\left(\mathrm{GroupOrder},L\right)$
 $\left[{50000}{,}{20000}\right]$ (21)
 > $L≔\mathrm{MinimalNormalSubgroups}\left(\mathrm{CyclicGroup}\left(100000\right)\right):$
 > $\mathrm{map}\left(\mathrm{GroupOrder},L\right)$
 $\left[{5}{,}{2}\right]$ (22)
 > $G≔\mathrm{CayleyTableGroup}\left(\mathrm{Symm}\left(3\right)\right)$
 ${G}{≔}{\mathrm{< a Cayley table group with 6 elements >}}$ (23)
 > $\mathrm{NormalSubgroups}\left(G\right)$
 $\left[{\mathrm{< a Cayley table group with 1 element >}}{,}{\mathrm{< a Cayley table group with 3 elements >}}{,}{\mathrm{< a Cayley table group with 6 elements >}}\right]$ (24)

Compatibility

 • The GroupTheory[NormalSubgroups] command was introduced in Maple 2016.
 • For more information on Maple 2016 changes, see Updates in Maple 2016.

 See Also