GroupTheory/EARNS - Maple Help

GroupTheory

 EARNS
 compute an elementary abelian regular normal subgroup of a primitive permutation group

 Calling Sequence EARNS( G )

Parameters

 G - PermutationGroup; a permutation group

Description

 • For a permutation group $G$, an "EARNS" is a normal subgroup of $G$ that is elementary abelian and acts regularly on the domain of action of $G$. A permutation group may, or may not, possess an EARNS.
 • The EARNS( G ) command returns an EARNS for a permutation group G, provided that one exists, and returns FAIL if G has no EARNS.
 • It is clear that for a permutation group to posess an EARNS it must be transitive and its support must have prime power cardinality. Therefore, EARNS returns FAIL if either of these conditions is not true.
 • In general, for Maple to identify an EARNS for a permutation group the group must either be primitive or a Frobenius group (or both). If G is neither primitive nor a Frobenius group, then EARNS may raise an exception indicating that the group is imprimitive and that Maple cannot, in that case, determine whether or not G has an EARNS.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Symm}\left(3\right)$
 ${G}{≔}{{\mathbf{S}}}_{{3}}$ (1)
 > $\mathrm{EARNS}\left(G\right)$
 $⟨\left({1}{,}{2}{,}{3}\right)⟩$ (2)
 > $\mathrm{EARNS}\left(\mathrm{Alt}\left(4\right)\right)$
 $⟨\left({1}{,}{3}\right)\left({2}{,}{4}\right){,}\left({1}{,}{2}\right)\left({3}{,}{4}\right)⟩$ (3)
 > $G≔\mathrm{Symm}\left(4\right)$
 ${G}{≔}{{\mathbf{S}}}_{{4}}$ (4)
 > $E≔\mathrm{EARNS}\left(G\right)$
 ${E}{≔}⟨\left({1}{,}{4}\right)\left({2}{,}{3}\right){,}\left({1}{,}{2}\right)\left({3}{,}{4}\right)⟩$ (5)
 > $\mathrm{IsNormal}\left(E,G\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IsRegular}\left(E\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsElementary}\left(E\right)$
 ${\mathrm{true}}$ (8)

A group acting on a set not of prime power cardinality can have no EARNS.

 > $G≔\mathrm{CyclicGroup}\left(10\right)$
 ${G}{≔}{{C}}_{{10}}$ (9)
 > $\mathrm{EARNS}\left(G\right)$
 ${\mathrm{FAIL}}$ (10)
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{true}}$ (11)
 > $\mathrm{type}\left(\mathrm{SupportLength}\left(G\right),'\mathrm{primepower}'\right)$
 ${\mathrm{false}}$ (12)

An intransitive group cannot posess an EARNS.

 > $G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1,2,3\right],\left[4,5\right]\right]\right)\right]\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩$ (13)
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{false}}$ (14)
 > $\mathrm{EARNS}\left(G\right)$
 ${\mathrm{FAIL}}$ (15)
 > $G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2,7,4,8,6,5,3\right]\right]\right),\mathrm{Perm}\left(\left[\left[2,4,3\right],\left[6,8,7\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,4\right],\left[5,6\right],\left[7,8\right]\right]\right)\right]\right)$
 ${G}{≔}⟨\left({2}{,}{7}{,}{4}{,}{8}{,}{6}{,}{5}{,}{3}\right){,}\left({2}{,}{4}{,}{3}\right)\left({6}{,}{8}{,}{7}\right){,}\left({1}{,}{2}\right)\left({3}{,}{4}\right)\left({5}{,}{6}\right)\left({7}{,}{8}\right)⟩$ (16)
 > $E≔\mathrm{EARNS}\left(G\right)$
 ${E}{≔}⟨\left({1}{,}{4}\right)\left({2}{,}{3}\right)\left({5}{,}{8}\right)\left({6}{,}{7}\right){,}\left({1}{,}{7}\right)\left({2}{,}{8}\right)\left({3}{,}{5}\right)\left({4}{,}{6}\right){,}\left({1}{,}{2}\right)\left({3}{,}{4}\right)\left({5}{,}{6}\right)\left({7}{,}{8}\right)⟩$ (17)

Primitive Frobenius groups always have an EARNS, the Frobenius kernel.

 > $G≔\mathrm{FrobeniusGroup}\left(14520,2\right)$
 ${G}{≔}⟨\left({2}{,}{6}{,}{74}{,}{80}{,}{16}\right)\left({3}{,}{49}{,}{63}{,}{116}{,}{19}\right)\left({4}{,}{51}{,}{108}{,}{68}{,}{59}\right)\left({5}{,}{53}{,}{52}{,}{110}{,}{40}\right)\left({7}{,}{10}{,}{37}{,}{47}{,}{78}\right)\left({8}{,}{17}{,}{61}{,}{60}{,}{90}\right)\left({9}{,}{20}{,}{89}{,}{97}{,}{93}\right)\left({11}{,}{43}{,}{106}{,}{48}{,}{31}\right)\left({12}{,}{103}{,}{70}{,}{29}{,}{36}\right)\left({13}{,}{104}{,}{85}{,}{86}{,}{84}\right)\left({14}{,}{81}{,}{38}{,}{42}{,}{41}\right)\left({15}{,}{58}{,}{64}{,}{79}{,}{77}\right)\left({18}{,}{105}{,}{56}{,}{101}{,}{91}\right)\left({21}{,}{102}{,}{109}{,}{54}{,}{112}\right)\left({22}{,}{26}{,}{100}{,}{55}{,}{32}\right)\left({23}{,}{39}{,}{28}{,}{33}{,}{35}\right)\left({24}{,}{83}{,}{98}{,}{71}{,}{75}\right)\left({25}{,}{118}{,}{76}{,}{119}{,}{95}\right)\left({27}{,}{30}{,}{117}{,}{94}{,}{92}\right)\left({34}{,}{99}{,}{120}{,}{69}{,}{72}\right)\left({44}{,}{62}{,}{113}{,}{87}{,}{82}\right)\left({45}{,}{121}{,}{88}{,}{66}{,}{57}\right)\left({46}{,}{115}{,}{107}{,}{114}{,}{65}\right)\left({50}{,}{111}{,}{67}{,}{96}{,}{73}\right){,}\left({2}{,}{22}{,}{21}{,}{99}{,}{104}\right)\left({3}{,}{58}{,}{75}{,}{73}{,}{20}\right)\left({4}{,}{26}{,}{19}{,}{65}{,}{78}\right)\left({5}{,}{64}{,}{16}{,}{92}{,}{36}\right)\left({6}{,}{9}{,}{91}{,}{37}{,}{17}\right)\left({7}{,}{94}{,}{85}{,}{96}{,}{106}\right)\left({8}{,}{52}{,}{54}{,}{57}{,}{63}\right)\left({10}{,}{23}{,}{102}{,}{15}{,}{38}\right)\left({11}{,}{95}{,}{93}{,}{87}{,}{109}\right)\left({12}{,}{114}{,}{89}{,}{34}{,}{39}\right)\left({13}{,}{119}{,}{70}{,}{81}{,}{49}\right)\left({14}{,}{108}{,}{98}{,}{113}{,}{74}\right)\left({18}{,}{84}{,}{45}{,}{83}{,}{35}\right)\left({24}{,}{32}{,}{61}{,}{103}{,}{43}\right)\left({25}{,}{33}{,}{68}{,}{90}{,}{30}\right)\left({27}{,}{116}{,}{62}{,}{69}{,}{56}\right)\left({28}{,}{67}{,}{44}{,}{100}{,}{40}\right)\left({29}{,}{101}{,}{50}{,}{112}{,}{51}\right)\left({31}{,}{110}{,}{41}{,}{115}{,}{105}\right)\left({42}{,}{55}{,}{121}{,}{97}{,}{117}\right)\left({46}{,}{80}{,}{66}{,}{111}{,}{118}\right)\left({47}{,}{76}{,}{120}{,}{71}{,}{53}\right)\left({48}{,}{72}{,}{88}{,}{79}{,}{59}\right)\left({60}{,}{77}{,}{82}{,}{86}{,}{107}\right){,}\left({2}{,}{3}\right)\left({4}{,}{5}\right)\left({6}{,}{49}\right)\left({7}{,}{12}\right)\left({8}{,}{14}\right)\left({9}{,}{13}\right)\left({10}{,}{103}\right)\left({11}{,}{35}\right)\left({15}{,}{32}\right)\left({16}{,}{19}\right)\left({17}{,}{81}\right)\left({18}{,}{95}\right)\left({20}{,}{104}\right)\left({21}{,}{75}\right)\left({22}{,}{58}\right)\left({23}{,}{43}\right)\left({24}{,}{102}\right)\left({25}{,}{105}\right)\left({26}{,}{64}\right)\left({27}{,}{46}\right)\left({28}{,}{48}\right)\left({29}{,}{47}\right)\left({30}{,}{115}\right)\left({31}{,}{33}\right)\left({34}{,}{96}\right)\left({36}{,}{78}\right)\left({37}{,}{70}\right)\left({38}{,}{61}\right)\left({39}{,}{106}\right)\left({40}{,}{59}\right)\left({41}{,}{90}\right)\left({42}{,}{60}\right)\left({44}{,}{88}\right)\left({45}{,}{87}\right)\left({50}{,}{120}\right)\left({51}{,}{53}\right)\left({52}{,}{108}\right)\left({54}{,}{98}\right)\left({55}{,}{77}\right)\left({56}{,}{118}\right)\left({57}{,}{113}\right)\left({62}{,}{66}\right)\left({63}{,}{74}\right)\left({65}{,}{92}\right)\left({67}{,}{72}\right)\left({68}{,}{110}\right)\left({69}{,}{111}\right)\left({71}{,}{112}\right)\left({73}{,}{99}\right)\left({76}{,}{101}\right)\left({79}{,}{100}\right)\left({80}{,}{116}\right)\left({82}{,}{121}\right)\left({83}{,}{109}\right)\left({84}{,}{93}\right)\left({85}{,}{89}\right)\left({86}{,}{97}\right)\left({91}{,}{119}\right)\left({94}{,}{114}\right)\left({107}{,}{117}\right){,}\left({1}{,}{2}{,}{7}{,}{27}{,}{79}{,}{53}{,}{51}{,}{100}{,}{46}{,}{12}{,}{3}\right)\left({4}{,}{8}{,}{28}{,}{72}{,}{52}{,}{84}{,}{70}{,}{112}{,}{102}{,}{47}{,}{13}\right)\left({5}{,}{9}{,}{29}{,}{24}{,}{71}{,}{37}{,}{93}{,}{108}{,}{67}{,}{48}{,}{14}\right)\left({6}{,}{22}{,}{69}{,}{97}{,}{66}{,}{20}{,}{17}{,}{15}{,}{19}{,}{35}{,}{23}\right)\left({10}{,}{38}{,}{94}{,}{107}{,}{50}{,}{36}{,}{33}{,}{87}{,}{76}{,}{82}{,}{39}\right)\left({11}{,}{16}{,}{32}{,}{81}{,}{104}{,}{62}{,}{86}{,}{111}{,}{58}{,}{49}{,}{43}\right)\left({18}{,}{59}{,}{75}{,}{110}{,}{105}{,}{99}{,}{44}{,}{26}{,}{77}{,}{113}{,}{60}\right)\left({21}{,}{40}{,}{95}{,}{42}{,}{57}{,}{55}{,}{64}{,}{88}{,}{73}{,}{25}{,}{68}\right)\left({30}{,}{85}{,}{65}{,}{116}{,}{118}{,}{83}{,}{34}{,}{90}{,}{98}{,}{119}{,}{74}\right)\left({31}{,}{78}{,}{120}{,}{117}{,}{114}{,}{61}{,}{103}{,}{106}{,}{121}{,}{101}{,}{45}\right)\left({41}{,}{96}{,}{109}{,}{56}{,}{80}{,}{92}{,}{89}{,}{115}{,}{63}{,}{91}{,}{54}\right){,}\left({1}{,}{4}{,}{16}{,}{26}{,}{78}{,}{65}{,}{92}{,}{36}{,}{64}{,}{19}{,}{5}\right)\left({2}{,}{8}{,}{32}{,}{77}{,}{120}{,}{116}{,}{89}{,}{33}{,}{88}{,}{35}{,}{9}\right)\left({3}{,}{13}{,}{11}{,}{44}{,}{31}{,}{85}{,}{80}{,}{50}{,}{55}{,}{15}{,}{14}\right)\left({6}{,}{24}{,}{27}{,}{72}{,}{104}{,}{60}{,}{114}{,}{83}{,}{63}{,}{76}{,}{25}\right)\left({7}{,}{28}{,}{81}{,}{113}{,}{117}{,}{118}{,}{115}{,}{87}{,}{73}{,}{23}{,}{29}\right)\left({10}{,}{40}{,}{97}{,}{93}{,}{51}{,}{70}{,}{111}{,}{75}{,}{106}{,}{98}{,}{41}\right)\left({12}{,}{47}{,}{43}{,}{99}{,}{45}{,}{30}{,}{56}{,}{107}{,}{57}{,}{17}{,}{48}\right)\left({18}{,}{61}{,}{34}{,}{91}{,}{82}{,}{68}{,}{22}{,}{71}{,}{79}{,}{52}{,}{62}\right)\left({20}{,}{67}{,}{46}{,}{102}{,}{49}{,}{105}{,}{101}{,}{74}{,}{109}{,}{94}{,}{42}\right)\left({21}{,}{69}{,}{37}{,}{53}{,}{84}{,}{86}{,}{59}{,}{103}{,}{90}{,}{54}{,}{39}\right)\left({38}{,}{95}{,}{66}{,}{108}{,}{100}{,}{112}{,}{58}{,}{110}{,}{121}{,}{119}{,}{96}\right)⟩$ (18)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{true}}$ (19)
 > $E≔\mathrm{EARNS}\left(G\right):$
 > $\mathrm{AreIsomorphic}\left(E,\mathrm{ElementaryGroup}\left(11,2\right)\right)$
 ${\mathrm{true}}$ (20)
 > $\mathrm{IsSubgroup}\left(E,\mathrm{FrobeniusKernel}\left(G\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{IsSubgroup}\left(\mathrm{FrobeniusKernel}\left(G\right),E\right)$
 ${\mathrm{true}}$ (21)
 > $G≔\mathrm{Symm}\left(2048\right)$
 ${G}{≔}{{\mathbf{S}}}_{{2048}}$ (22)
 > $\mathrm{EARNS}\left(G\right)$
 ${\mathrm{FAIL}}$ (23)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{true}}$ (24)
 > $G≔\mathrm{Alt}\left({5}^{5}\right)$
 ${G}{≔}{{\mathbf{A}}}_{{3125}}$ (25)
 > $\mathrm{EARNS}\left(G\right)$
 ${\mathrm{FAIL}}$ (26)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{true}}$ (27)

A regular elementary abelian transitive group is its own EARNS, even if it does not act primitively.

 > $G≔\mathrm{TransitiveGroup}\left(27,4\right):$
 > $\mathrm{EARNS}\left(G\right)$
 $⟨\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}{,}{6}\right)\left({7}{,}{8}{,}{9}\right)\left({10}{,}{11}{,}{12}\right)\left({13}{,}{14}{,}{15}\right)\left({16}{,}{17}{,}{18}\right)\left({19}{,}{20}{,}{21}\right)\left({22}{,}{23}{,}{24}\right)\left({25}{,}{26}{,}{27}\right){,}\left({1}{,}{6}{,}{26}\right)\left({2}{,}{4}{,}{27}\right)\left({3}{,}{5}{,}{25}\right)\left({7}{,}{10}{,}{13}\right)\left({8}{,}{11}{,}{14}\right)\left({9}{,}{12}{,}{15}\right)\left({16}{,}{21}{,}{23}\right)\left({17}{,}{19}{,}{24}\right)\left({18}{,}{20}{,}{22}\right){,}\left({1}{,}{11}{,}{19}\right)\left({2}{,}{12}{,}{20}\right)\left({3}{,}{10}{,}{21}\right)\left({4}{,}{15}{,}{22}\right)\left({5}{,}{13}{,}{23}\right)\left({6}{,}{14}{,}{24}\right)\left({7}{,}{16}{,}{25}\right)\left({8}{,}{17}{,}{26}\right)\left({9}{,}{18}{,}{27}\right)⟩$ (28)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{false}}$ (29)
 > $\mathrm{IsRegular}\left(G\right)$
 ${\mathrm{true}}$ (30)

Some imprimitive Frobenius groups have an EARNS.

 > $G≔\mathrm{TransitiveGroup}\left(25,9\right)$
 ${G}{≔}⟨\left({2}{,}{3}{,}{5}{,}{4}\right)\left({6}{,}{14}{,}{24}{,}{17}\right)\left({7}{,}{11}{,}{23}{,}{20}\right)\left({8}{,}{13}{,}{22}{,}{18}\right)\left({9}{,}{15}{,}{21}{,}{16}\right)\left({10}{,}{12}{,}{25}{,}{19}\right){,}\left({1}{,}{7}{,}{11}{,}{20}{,}{23}\right)\left({2}{,}{8}{,}{12}{,}{16}{,}{24}\right)\left({3}{,}{9}{,}{13}{,}{17}{,}{25}\right)\left({4}{,}{10}{,}{14}{,}{18}{,}{21}\right)\left({5}{,}{6}{,}{15}{,}{19}{,}{22}\right){,}\left({1}{,}{25}{,}{19}{,}{12}{,}{10}\right)\left({2}{,}{21}{,}{20}{,}{13}{,}{6}\right)\left({3}{,}{22}{,}{16}{,}{14}{,}{7}\right)\left({4}{,}{23}{,}{17}{,}{15}{,}{8}\right)\left({5}{,}{24}{,}{18}{,}{11}{,}{9}\right)⟩$ (31)
 > $\mathrm{EARNS}\left(G\right)$
 ${Fitt}{}\left(⟨\left({2}{,}{3}{,}{5}{,}{4}\right)\left({6}{,}{14}{,}{24}{,}{17}\right)\left({7}{,}{11}{,}{23}{,}{20}\right)\left({8}{,}{13}{,}{22}{,}{18}\right)\left({9}{,}{15}{,}{21}{,}{16}\right)\left({10}{,}{12}{,}{25}{,}{19}\right){,}\left({1}{,}{7}{,}{11}{,}{20}{,}{23}\right)\left({2}{,}{8}{,}{12}{,}{16}{,}{24}\right)\left({3}{,}{9}{,}{13}{,}{17}{,}{25}\right)\left({4}{,}{10}{,}{14}{,}{18}{,}{21}\right)\left({5}{,}{6}{,}{15}{,}{19}{,}{22}\right){,}\left({1}{,}{25}{,}{19}{,}{12}{,}{10}\right)\left({2}{,}{21}{,}{20}{,}{13}{,}{6}\right)\left({3}{,}{22}{,}{16}{,}{14}{,}{7}\right)\left({4}{,}{23}{,}{17}{,}{15}{,}{8}\right)\left({5}{,}{24}{,}{18}{,}{11}{,}{9}\right)⟩\right)$ (32)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{false}}$ (33)

But not all do.

 > $G≔\mathrm{TransitiveGroup}\left(16,63\right)$
 ${G}{≔}⟨\left({2}{,}{3}{,}{4}\right)\left({5}{,}{15}{,}{9}\right)\left({6}{,}{13}{,}{12}\right)\left({7}{,}{14}{,}{10}\right)\left({8}{,}{16}{,}{11}\right){,}\left({1}{,}{5}{,}{3}{,}{7}\right)\left({2}{,}{6}{,}{4}{,}{8}\right)\left({9}{,}{14}{,}{11}{,}{16}\right)\left({10}{,}{13}{,}{12}{,}{15}\right)⟩$ (34)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{false}}$ (35)
 > $\mathrm{IsFrobeniusPermGroup}\left(G\right)$
 ${\mathrm{true}}$ (36)
 > $\mathrm{EARNS}\left(G\right)$
 ${\mathrm{FAIL}}$ (37)

In most cases, however, an exception is raised if the input to EARNS is imprimitive.

 > $G≔\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(3\right),\mathrm{CyclicGroup}\left(3\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}\right){,}\left({1}{,}{4}{,}{7}\right)\left({2}{,}{5}{,}{8}\right)\left({3}{,}{6}{,}{9}\right)⟩$ (38)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{false}}$ (39)
 > $\mathrm{EARNS}\left(G\right)$