GroupTheory/IsFrobeniusGroup - Maple Help

GroupTheory

 IsFrobeniusPermGroup
 determine whether a group is a Frobenius permutation group
 IsFrobeniusGroup
 determine whether a group is a Frobenius group
 FrobeniusKernel
 compute the Frobenius kernel of a Frobenius group
 FrobeniusComplement
 compute the Frobenius complement of a Frobenius group
 FrobeniusPermRep
 compute a Frobenius permutation group isomorphic to a given Frobenius group

 Calling Sequence IsFrobeniusPermGroup( G ) IsFrobeniusGroup( G ) FrobeniusKernel( G ) FrobeniusComplement( G ) FrobeniusPermRep( G )

Parameters

 G - a permutation group

Description

 • A permutation group $G$ is a Frobenius group if it is transitive, has a non-trivial point stabilizer, and no non-trivial element of $G$ fixes more than one point.
 • The IsFrobeniusPermGroup( G ) command returns true if the permutation group G is a Frobenius group, and returns false otherwise.
 • An abstract group $G$ is a Frobenius group if it has a proper, non-trivial malnormal subgroup self-centralising subgroup $H$, called a Frobenius complement. In this case, $G$ has a normal (even characteristic) subgroup $K$, called the Frobenius kernel, consisting of the identity element of $G$ and the elements of $G$ that do not belong to any conjugate of $H$ in $G$.
 • The IsFrobeniusGroup( G ) command returns true if G is a Frobenius group as an abstract group, and returns false otherwise.
 • The two definitions are equivalent in the following sense.  If $G$ is a Frobenius permutation group, then $G$ is Frobenius as an abstract group, with the stabilizer of a point being a Frobenius complement in $G$. Conversely, if $G$ is Frobenius as an abstract group, then the action of $G$ on the cosets of a Frobenius complement is faithful and is Frobenius as a permutation group, and so $G$ is isomorphic to the corresponding Frobenius permutation group,
 • The Frobenius kernel of a Frobenius group $G$ is uniquely defined, because a group can be a Frobenius group in at most one way. The Frobenius complement of a Frobenius group $G$ is well-defined up to conjugacy in $G$.
 • If $G$ is a Frobenius group, the FrobeniusKernel( G ) command returns the Frobenius kernel of $G$.  If $G$ is not Frobenius, an exception is raised.
 • If $G$ is a Frobenius group, the FrobeniusComplement( G ) command returns a Frobenius complement of $G$.  If $G$ is not Frobenius, an exception is raised.
 • For a Frobenius group G, the FrobeniusPermRep( G ) command returns a Frobenius permutation group isomorphic to $G$. It is permutation isomorphic to the action on $G$ on the cosets of a Frobenius complement in $G$.

Examples

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

The smallest Frobenius group is the symmetric group of degree $3$.

 > $\mathrm{IsFrobeniusGroup}\left(\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{IsFrobeniusPermGroup}\left(\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{FrobeniusKernel}\left(\mathrm{Symm}\left(3\right)\right)$
 $⟨\left({1}{,}{2}{,}{3}\right)⟩$ (3)
 > $\mathrm{FrobeniusComplement}\left(\mathrm{Symm}\left(3\right)\right)$
 $⟨\left({1}{,}{2}\right)⟩$ (4)
 > $\mathrm{IsMalnormal}\left(\mathrm{FrobeniusComplement}\left(\mathrm{Symm}\left(3\right)\right),\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (5)

A different permutation group isomorphic to the symmetric group of degree $3$ is a Frobenius group, but is not Frobenius as a permutation group.

 > $G≔\mathrm{SmallGroup}\left(6,1\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right){,}\left({1}{,}{3}{,}{4}\right)\left({2}{,}{5}{,}{6}\right)⟩$ (6)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsFrobeniusGroup}\left(G\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{IsFrobeniusPermGroup}\left(G\right)$
 ${\mathrm{false}}$ (9)
 > $\mathrm{FrobeniusKernel}\left(G\right)$
 $⟨\left({1}{,}{4}{,}{3}\right)\left({2}{,}{6}{,}{5}\right)⟩$ (10)
 > $\mathrm{FrobeniusComplement}\left(G\right)$
 $⟨\left({1}{,}{2}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right)⟩$ (11)
 > $H≔\mathrm{FrobeniusPermRep}\left(G\right)$
 ${H}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{3}{,}{2}\right)⟩$ (12)
 > $\mathrm{IsFrobeniusPermGroup}\left(H\right)$
 ${\mathrm{true}}$ (13)
 > $\mathrm{AreIsomorphic}\left(H,G\right)$
 ${\mathrm{true}}$ (14)

The dihedral group ${\mathrm{D}}_{n}$ is Frobenius if, and only, if, $n$ is odd.

 > $\mathrm{IsFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{false}}$ (15)
 > $\mathrm{IsFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(5\right)\right)$
 ${\mathrm{true}}$ (16)
 > $\mathrm{IsFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(6\right)\right)$
 ${\mathrm{false}}$ (17)
 > $\mathrm{IsFrobeniusGroup}\left(\mathrm{PSL}\left(2,3\right)\right)$
 ${\mathrm{true}}$ (18)

We construct here a Frobenius subgroup of order $110$ in the first Janko group.

 > $a,b≔\mathrm{op}\left(\mathrm{Generators}\left(\mathrm{JankoGroup}\left(1\right)\right)\right):$
 > $u≔{\left(a·b\right)}^{-2}·b·a·b:$$\mathrm{PermOrder}\left(u\right)$
 ${2}$ (19)
 > $v≔{\left(a·b·b\right)}^{-2}·{\left(a·b·a·b·a·b·b·a·b·a·b·b\right)}^{3}·{\left(a·b·b\right)}^{2}:$$\mathrm{PermOrder}\left(v\right)$
 ${5}$ (20)
 > $G≔\mathrm{Group}\left(\left[u,v\right]\right):$$\mathrm{GroupOrder}\left(G\right)$
 ${110}$ (21)
 > $\mathrm{IsFrobeniusGroup}\left(G\right)$
 ${\mathrm{true}}$ (22)

However, this is not a Frobenius action; to get a Frobenius permutation group, use FrobeniusPermRep.

 > $\mathrm{IsFrobeniusPermGroup}\left(G\right)$
 ${\mathrm{false}}$ (23)
 > $P≔\mathrm{FrobeniusPermRep}\left(G\right)$
 ${P}{≔}⟨\left({1}{,}{8}\right)\left({2}{,}{9}\right)\left({3}{,}{5}\right)\left({6}{,}{10}\right)\left({7}{,}{11}\right){,}\left({1}{,}{11}{,}{10}{,}{7}{,}{9}\right)\left({3}{,}{4}{,}{6}{,}{8}{,}{5}\right)⟩$ (24)
 > $\mathrm{IsFrobeniusPermGroup}\left(P\right)$
 ${\mathrm{true}}$ (25)

Now we can compute the Frobenius kernel and complement, and determine their orders.

 > $K≔\mathrm{FrobeniusKernel}\left(P\right)$
 ${K}{≔}⟨\left({1}{,}{10}{,}{11}{,}{5}{,}{9}{,}{2}{,}{3}{,}{7}{,}{6}{,}{8}{,}{4}\right)⟩$ (26)
 > $\mathrm{GroupOrder}\left(K\right)$
 ${11}$ (27)
 > $C≔\mathrm{FrobeniusComplement}\left(P\right)$
 ${C}{≔}{\mathrm{< a permutation group on 11 letters with 6 generators >}}$ (28)
 > $\mathrm{GroupOrder}\left(C\right)$
 ${10}$ (29)
 > $\mathrm{IsMalnormal}\left(C,P\right)$
 ${\mathrm{true}}$ (30)

Of course, we obtain the same result by computing the Frobenius kernel and complement of G itself.

 > $K≔\mathrm{FrobeniusKernel}\left(G\right)$
 ${K}{≔}⟨\left({1}{,}{164}{,}{50}{,}{38}{,}{76}{,}{254}{,}{79}{,}{34}{,}{89}{,}{85}{,}{142}\right)\left({2}{,}{25}{,}{150}{,}{151}{,}{158}{,}{80}{,}{86}{,}{260}{,}{265}{,}{214}{,}{127}\right)\left({3}{,}{237}{,}{11}{,}{67}{,}{95}{,}{181}{,}{81}{,}{174}{,}{186}{,}{98}{,}{96}\right)\left({4}{,}{216}{,}{52}{,}{190}{,}{48}{,}{198}{,}{201}{,}{148}{,}{126}{,}{195}{,}{176}\right)\left({5}{,}{211}{,}{84}{,}{19}{,}{61}{,}{99}{,}{212}{,}{53}{,}{75}{,}{235}{,}{226}\right)\left({6}{,}{185}{,}{106}{,}{88}{,}{123}{,}{24}{,}{69}{,}{108}{,}{60}{,}{132}{,}{28}\right)\left({7}{,}{125}{,}{177}{,}{29}{,}{15}{,}{217}{,}{184}{,}{131}{,}{22}{,}{37}{,}{43}\right)\left({8}{,}{59}{,}{104}{,}{90}{,}{191}{,}{244}{,}{56}{,}{138}{,}{163}{,}{116}{,}{243}\right)\left({9}{,}{156}{,}{135}{,}{197}{,}{239}{,}{207}{,}{236}{,}{266}{,}{249}{,}{241}{,}{145}\right)\left({10}{,}{210}{,}{193}{,}{157}{,}{188}{,}{105}{,}{82}{,}{240}{,}{17}{,}{12}{,}{133}\right)\left({13}{,}{44}{,}{255}{,}{258}{,}{21}{,}{107}{,}{153}{,}{72}{,}{65}{,}{173}{,}{209}\right)\left({14}{,}{225}{,}{152}{,}{162}{,}{221}{,}{46}{,}{18}{,}{41}{,}{63}{,}{57}{,}{141}\right)\left({16}{,}{238}{,}{262}{,}{223}{,}{33}{,}{26}{,}{39}{,}{77}{,}{222}{,}{36}{,}{218}\right)\left({20}{,}{172}{,}{261}{,}{220}{,}{180}{,}{27}{,}{263}{,}{202}{,}{118}{,}{47}{,}{250}\right)\left({23}{,}{224}{,}{154}{,}{256}{,}{58}{,}{92}{,}{143}{,}{252}{,}{208}{,}{187}{,}{182}\right)\left({30}{,}{70}{,}{119}{,}{97}{,}{113}{,}{245}{,}{253}{,}{169}{,}{124}{,}{74}{,}{54}\right)\left({31}{,}{87}{,}{149}{,}{242}{,}{45}{,}{83}{,}{51}{,}{170}{,}{35}{,}{167}{,}{100}\right)\left({32}{,}{178}{,}{109}{,}{246}{,}{248}{,}{229}{,}{192}{,}{194}{,}{228}{,}{42}{,}{166}\right)\left({40}{,}{134}{,}{130}{,}{200}{,}{168}{,}{146}{,}{64}{,}{230}{,}{179}{,}{165}{,}{159}\right)\left({49}{,}{139}{,}{264}{,}{219}{,}{234}{,}{155}{,}{94}{,}{102}{,}{78}{,}{215}{,}{175}\right)\left({55}{,}{199}{,}{144}{,}{115}{,}{110}{,}{189}{,}{231}{,}{183}{,}{251}{,}{171}{,}{91}\right)\left({62}{,}{257}{,}{247}{,}{114}{,}{112}{,}{101}{,}{71}{,}{68}{,}{73}{,}{232}{,}{259}\right)\left({66}{,}{122}{,}{121}{,}{196}{,}{213}{,}{120}{,}{136}{,}{111}{,}{160}{,}{161}{,}{140}\right)\left({93}{,}{206}{,}{203}{,}{205}{,}{233}{,}{137}{,}{227}{,}{204}{,}{103}{,}{128}{,}{117}\right)⟩$ (31)
 > $\mathrm{GroupOrder}\left(K\right)$
 ${11}$ (32)
 > $C≔\mathrm{FrobeniusComplement}\left(G\right)$
 ${C}{≔}⟨\left({1}{,}{77}{,}{247}{,}{109}{,}{135}\right)\left({2}{,}{170}{,}{237}{,}{74}{,}{118}\right)\left({3}{,}{113}{,}{27}{,}{260}{,}{167}\right)\left({4}{,}{78}{,}{57}{,}{72}{,}{37}\right)\left({5}{,}{179}{,}{213}{,}{12}{,}{69}\right)\left({6}{,}{19}{,}{134}{,}{121}{,}{133}\right)\left({7}{,}{176}{,}{219}{,}{18}{,}{258}\right)\left({8}{,}{138}{,}{116}{,}{163}{,}{90}\right)\left({9}{,}{164}{,}{238}{,}{101}{,}{192}\right)\left({10}{,}{24}{,}{212}{,}{146}{,}{66}\right)\left({11}{,}{97}{,}{20}{,}{158}{,}{83}\right)\left({13}{,}{131}{,}{216}{,}{264}{,}{225}\right)\left({14}{,}{153}{,}{15}{,}{126}{,}{234}\right)\left({16}{,}{62}{,}{246}{,}{266}{,}{38}\right)\left({17}{,}{185}{,}{75}{,}{168}{,}{136}\right)\left({21}{,}{217}{,}{52}{,}{102}{,}{221}\right)\left({22}{,}{201}{,}{155}{,}{46}{,}{173}\right)\left({23}{,}{58}{,}{154}{,}{256}{,}{208}\right)\left({25}{,}{87}{,}{95}{,}{119}{,}{263}\right)\left({26}{,}{114}{,}{194}{,}{207}{,}{76}\right)\left({28}{,}{235}{,}{165}{,}{161}{,}{210}\right)\left({29}{,}{190}{,}{139}{,}{41}{,}{65}\right)\left({30}{,}{202}{,}{86}{,}{45}{,}{186}\right)\left({31}{,}{96}{,}{54}{,}{261}{,}{151}\right)\left({32}{,}{197}{,}{254}{,}{218}{,}{71}\right)\left({33}{,}{232}{,}{248}{,}{156}{,}{79}\right)\left({34}{,}{36}{,}{257}{,}{228}{,}{145}\right)\left({35}{,}{81}{,}{70}{,}{172}{,}{214}\right)\left({39}{,}{73}{,}{166}{,}{241}{,}{50}\right)\left({40}{,}{160}{,}{240}{,}{108}{,}{99}\right)\left({42}{,}{239}{,}{142}{,}{262}{,}{259}\right)\left({43}{,}{198}{,}{49}{,}{152}{,}{107}\right)\left({44}{,}{125}{,}{48}{,}{94}{,}{141}\right)\left({47}{,}{80}{,}{100}{,}{181}{,}{169}\right)\left({51}{,}{174}{,}{253}{,}{180}{,}{150}\right)\left({53}{,}{159}{,}{196}{,}{188}{,}{88}\right)\left({55}{,}{199}{,}{231}{,}{171}{,}{144}\right)\left({56}{,}{191}{,}{244}{,}{243}{,}{104}\right)\left({60}{,}{226}{,}{200}{,}{122}{,}{105}\right)\left({61}{,}{64}{,}{120}{,}{157}{,}{132}\right)\left({63}{,}{255}{,}{184}{,}{148}{,}{175}\right)\left({67}{,}{124}{,}{220}{,}{265}{,}{242}\right)\left({68}{,}{229}{,}{236}{,}{85}{,}{222}\right)\left({82}{,}{106}{,}{84}{,}{230}{,}{140}\right)\left({89}{,}{223}{,}{112}{,}{178}{,}{249}\right)\left({91}{,}{183}{,}{115}{,}{189}{,}{110}\right)\left({92}{,}{252}{,}{143}{,}{224}{,}{187}\right)\left({93}{,}{103}{,}{233}{,}{227}{,}{137}\right)\left({98}{,}{245}{,}{250}{,}{127}{,}{149}\right)\left({111}{,}{193}{,}{123}{,}{211}{,}{130}\right)\left({117}{,}{205}{,}{206}{,}{203}{,}{204}\right)\left({162}{,}{209}{,}{177}{,}{195}{,}{215}\right){,}\left({1}{,}{21}\right)\left({2}{,}{214}\right)\left({3}{,}{174}\right)\left({4}{,}{112}\right)\left({5}{,}{235}\right)\left({6}{,}{24}\right)\left({7}{,}{238}\right)\left({8}{,}{171}\right)\left({9}{,}{18}\right)\left({10}{,}{133}\right)\left({11}{,}{181}\right)\left({12}{,}{210}\right)\left({13}{,}{76}\right)\left({14}{,}{236}\right)\left({15}{,}{222}\right)\left({16}{,}{125}\right)\left({17}{,}{193}\right)\left({19}{,}{212}\right)\left({20}{,}{47}\right)\left({22}{,}{33}\right)\left({23}{,}{103}\right)\left({25}{,}{265}\right)\left({26}{,}{131}\right)\left({27}{,}{180}\right)\left({28}{,}{69}\right)\left({29}{,}{36}\right)\left({30}{,}{54}\right)\left({31}{,}{45}\right)\left({32}{,}{215}\right)\left({34}{,}{65}\right)\left({35}{,}{170}\right)\left({37}{,}{223}\right)\left({38}{,}{44}\right)\left({39}{,}{184}\right)\left({40}{,}{64}\right)\left({41}{,}{145}\right)\left({42}{,}{49}\right)\left({43}{,}{262}\right)\left({46}{,}{156}\right)\left({48}{,}{62}\right)\left({50}{,}{255}\right)\left({51}{,}{167}\right)\left({52}{,}{247}\right)\left({53}{,}{84}\right)\left({55}{,}{116}\right)\left({56}{,}{115}\right)\left({57}{,}{249}\right)\left({58}{,}{233}\right)\left({59}{,}{251}\right)\left({61}{,}{99}\right)\left({63}{,}{241}\right)\left({66}{,}{121}\right)\left({67}{,}{95}\right)\left({68}{,}{126}\right)\left({70}{,}{74}\right)\left({71}{,}{195}\right)\left({72}{,}{89}\right)\left({73}{,}{148}\right)\left({75}{,}{211}\right)\left({77}{,}{217}\right)\left({78}{,}{178}\right)\left({79}{,}{173}\right)\left({80}{,}{158}\right)\left({81}{,}{237}\right)\left({82}{,}{188}\right)\left({83}{,}{100}\right)\left({85}{,}{153}\right)\left({86}{,}{151}\right)\left({87}{,}{242}\right)\left({88}{,}{106}\right)\left({90}{,}{231}\right)\left({91}{,}{243}\right)\left({92}{,}{205}\right)\left({93}{,}{208}\right)\left({94}{,}{246}\right)\left({96}{,}{186}\right)\left({97}{,}{169}\right)\left({101}{,}{176}\right)\left({102}{,}{109}\right)\left({104}{,}{183}\right)\left({107}{,}{142}\right)\left({108}{,}{132}\right)\left({110}{,}{244}\right)\left({111}{,}{136}\right)\left({113}{,}{253}\right)\left({114}{,}{216}\right)\left({117}{,}{187}\right)\left({118}{,}{172}\right)\left({119}{,}{124}\right)\left({120}{,}{160}\right)\left({123}{,}{185}\right)\left({128}{,}{182}\right)\left({129}{,}{147}\right)\left({130}{,}{168}\right)\left({134}{,}{146}\right)\left({135}{,}{221}\right)\left({137}{,}{256}\right)\left({138}{,}{144}\right)\left({139}{,}{228}\right)\left({140}{,}{196}\right)\left({141}{,}{266}\right)\left({143}{,}{203}\right)\left({150}{,}{260}\right)\left({152}{,}{239}\right)\left({154}{,}{227}\right)\left({155}{,}{248}\right)\left({157}{,}{240}\right)\left({159}{,}{230}\right)\left({161}{,}{213}\right)\left({162}{,}{197}\right)\left({163}{,}{199}\right)\left({164}{,}{258}\right)\left({165}{,}{179}\right)\left({166}{,}{175}\right)\left({177}{,}{218}\right)\left({189}{,}{191}\right)\left({190}{,}{257}\right)\left({192}{,}{219}\right)\left({194}{,}{264}\right)\left({198}{,}{259}\right)\left({201}{,}{232}\right)\left({202}{,}{261}\right)\left({204}{,}{224}\right)\left({206}{,}{252}\right)\left({207}{,}{225}\right)\left({209}{,}{254}\right)\left({220}{,}{263}\right)\left({229}{,}{234}\right)⟩$ (33)
 > $\mathrm{GroupOrder}\left(C\right)$
 ${10}$ (34)
 > $\mathrm{IsMalnormal}\left(C,G\right)$
 ${\mathrm{true}}$ (35)

The Frobenius complement in a Frobenius dihedral group is a subgroup of order two.

 > $G≔\mathrm{DihedralGroup}\left(7\right):$
 > $C≔\mathrm{FrobeniusComplement}\left(G\right)$
 ${C}{≔}⟨\left({1}{,}{4}\right)\left({2}{,}{3}\right)\left({5}{,}{7}\right)⟩$ (36)
 > $\mathrm{IsMalnormal}\left(C,G\right)$
 ${\mathrm{true}}$ (37)

The Mathieu group of degree $10$ has a point stabiliser of order $72$. (This is sometimes referred to as a Mathieu group of degree $9$.)

 > $G≔\mathrm{MathieuGroup}\left(10\right)$
 ${G}{≔}{{M}}_{{10}}$ (38)
 > $S≔\mathrm{Stabiliser}\left(1,G\right)$
 ${S}{≔}{\mathrm{< a permutation group on 10 letters with 7 generators >}}$ (39)
 > $\mathrm{GroupOrder}\left(S\right)$
 ${72}$ (40)

This point stabiliser is a Frobenius group.

 > $\mathrm{IsFrobeniusGroup}\left(S\right)$
 ${\mathrm{true}}$ (41)

Moreover, the action is Frobenius.

 > $\mathrm{IsFrobeniusPermGroup}\left(S\right)$
 ${\mathrm{true}}$ (42)

The Frobenius complement in $S$ is a quaternion group.

 > $\mathrm{AreIsomorphic}\left(\mathrm{FrobeniusComplement}\left(S\right),\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (43)

Compatibility

 • The GroupTheory[IsFrobeniusPermGroup], GroupTheory[IsFrobeniusGroup], GroupTheory[FrobeniusKernel], GroupTheory[FrobeniusComplement] and GroupTheory[FrobeniusPermRep] commands were introduced in Maple 2019.