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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-centralizing 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.

 > $G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,6\right],\left[4,5\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,3,4\right],\left[2,5,6\right]\right]\right)\right]\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right){,}\left({1}{,}{3}{,}{4}\right)\left({2}{,}{5}{,}{6}\right)⟩$ (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)
 > $\mathrm{FrobeniusKernel}\left(G\right)$
 $⟨\left({1}{,}{3}{,}{4}\right)\left({2}{,}{5}{,}{6}\right)⟩$ (10)
 > $\mathrm{FrobeniusComplement}\left(G\right)$
 $⟨\left({1}{,}{6}\right)\left({2}{,}{4}\right)\left({3}{,}{5}\right)⟩$ (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)
 > $\mathrm{IsFrobeniusPermGroup}\left(\mathrm{AGL}\left(1,243\right)\right)$
 ${\mathrm{true}}$ (19)

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≔\mathrm{.}\left({\left(\mathrm{.}\left(a,b\right)\right)}^{-2},b,a,b\right):$$\mathrm{PermOrder}\left(u\right)$
 ${2}$ (20)
 > $v≔\mathrm{.}\left({\left(\mathrm{.}\left(a,b,b\right)\right)}^{-2},{\left(\mathrm{.}\left(a,b,a,b,a,b,b,a,b,a,b,b\right)\right)}^{3},{\left(\mathrm{.}\left(a,b,b\right)\right)}^{2}\right):$$\mathrm{PermOrder}\left(v\right)$
 ${5}$ (21)
 > $G≔\mathrm{Group}\left(\left[u,v\right]\right):$$\mathrm{GroupOrder}\left(G\right)$
 ${110}$ (22)
 > $\mathrm{IsFrobeniusGroup}\left(G\right)$
 ${\mathrm{true}}$ (23)

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

 > $\mathrm{IsFrobeniusPermGroup}\left(G\right)$
 ${\mathrm{false}}$ (24)
 > $P≔\mathrm{FrobeniusPermRep}\left(G\right)$
 ${P}{≔}⟨\left({1}{,}{7}\right)\left({2}{,}{11}\right)\left({3}{,}{6}\right)\left({5}{,}{9}\right)\left({8}{,}{10}\right){,}\left({1}{,}{11}{,}{8}{,}{5}{,}{10}\right)\left({2}{,}{3}{,}{6}{,}{4}{,}{9}\right)⟩$ (25)
 > $\mathrm{IsFrobeniusPermGroup}\left(P\right)$
 ${\mathrm{true}}$ (26)

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

 > $K≔\mathrm{FrobeniusKernel}\left(P\right)$
 ${K}{≔}⟨\left({1}{,}{9}{,}{5}{,}{7}{,}{2}{,}{8}{,}{6}{,}{4}{,}{3}{,}{10}{,}{11}\right)⟩$ (27)
 > $\mathrm{GroupOrder}\left(K\right)$
 ${11}$ (28)
 > $C≔\mathrm{FrobeniusComplement}\left(P\right)$
 ${C}{≔}⟨\left({2}{,}{9}{,}{7}{,}{10}{,}{8}\right)\left({3}{,}{5}{,}{6}{,}{4}{,}{11}\right){,}\left({2}{,}{5}{,}{9}{,}{6}{,}{7}{,}{4}{,}{10}{,}{11}{,}{8}{,}{3}\right)⟩$ (29)
 > $\mathrm{GroupOrder}\left(C\right)$
 ${10}$ (30)
 > $\mathrm{IsMalnormal}\left(C,P\right)$
 ${\mathrm{true}}$ (31)

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

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

 > $G≔\mathrm{DihedralGroup}\left(7\right):$
 > $C≔\mathrm{FrobeniusComplement}\left(G\right)$
 ${C}{≔}⟨\left({1}{,}{6}\right)\left({2}{,}{5}\right)\left({3}{,}{4}\right)⟩$ (37)
 > $\mathrm{IsMalnormal}\left(C,G\right)$
 ${\mathrm{true}}$ (38)

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

 > $G≔\mathrm{MathieuGroup}\left(10\right)$
 ${G}{≔}{{M}}_{{10}}$ (39)
 > $S≔\mathrm{Stabilizer}\left(1,G\right)$
 ${S}{≔}⟨\left({2}{,}{8}{,}{4}{,}{6}\right)\left({3}{,}{9}{,}{5}{,}{10}\right){,}\left({2}{,}{6}{,}{7}{,}{9}\right)\left({3}{,}{5}{,}{8}{,}{10}\right)⟩$ (40)
 > $\mathrm{GroupOrder}\left(S\right)$
 ${72}$ (41)

This point stabilizer is a Frobenius group.

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

Moreover, the action is Frobenius.

 > $\mathrm{IsFrobeniusPermGroup}\left(S\right)$
 ${\mathrm{true}}$ (43)

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

 > $\mathrm{AreIsomorphic}\left(\mathrm{FrobeniusComplement}\left(S\right),\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (44)

Compatibility

 • The GroupTheory[IsFrobeniusPermGroup], GroupTheory[IsFrobeniusGroup], GroupTheory[FrobeniusKernel], GroupTheory[FrobeniusComplement] and GroupTheory[FrobeniusPermRep] commands were introduced in Maple 2019.
 • For more information on Maple 2019 changes, see Updates in Maple 2019.