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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

Parameters

Description

Examples

Compatibility

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

withGroupTheory:

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

IsFrobeniusGroupSymm3

true

(1)

IsFrobeniusPermGroupSymm3

true

(2)

FrobeniusKernelSymm3

1,2,3

(3)

FrobeniusComplementSymm3

2,3

(4)

IsMalnormalFrobeniusComplementSymm3,Symm3

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.

GSmallGroup6,1

G1,23,64,5,1,3,42,5,6

(6)

AreIsomorphicG,Symm3

true

(7)

IsFrobeniusGroupG

true

(8)

IsFrobeniusPermGroupG

false

(9)

FrobeniusKernelG

1,4,32,6,5

(10)

FrobeniusComplementG

1,52,34,6

(11)

HFrobeniusPermRepG

H1,2,1,3,2

(12)

IsFrobeniusPermGroupH

true

(13)

AreIsomorphicH,G

true

(14)

The dihedral group Dn is Frobenius if, and only, if, n is odd.

IsFrobeniusGroupDihedralGroup4

false

(15)

IsFrobeniusGroupDihedralGroup5

true

(16)

IsFrobeniusGroupDihedralGroup6

false

(17)

IsFrobeniusGroupPSL2,3

true

(18)

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

a,bopGeneratorsJankoGroup1:

ua·b2·b·a·b:PermOrderu

2

(19)

va·b·b2·a·b·a·b·a·b·b·a·b·a·b·b3·a·b·b2:PermOrderv

5

(20)

GGroupu,v:GroupOrderG

110

(21)

IsFrobeniusGroupG

true

(22)

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

IsFrobeniusPermGroupG

false

(23)

PFrobeniusPermRepG

P1,62,93,84,57,10,1,5,6,8,24,9,10,7,11

(24)

IsFrobeniusPermGroupP

true

(25)

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

KFrobeniusKernelP

K1,10,8,3,7,6,4,9,11,2,5

(26)

GroupOrderK

11

(27)

CFrobeniusComplementP

C < a permutation group on 11 letters with 6 generators >

(28)

GroupOrderC

10

(29)

IsMalnormalC&comma;P

true

(30)

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

KFrobeniusKernelG

K1&comma;50&comma;76&comma;79&comma;89&comma;142&comma;164&comma;38&comma;254&comma;34&comma;852&comma;150&comma;158&comma;86&comma;265&comma;127&comma;25&comma;151&comma;80&comma;260&comma;2143&comma;11&comma;95&comma;81&comma;186&comma;96&comma;237&comma;67&comma;181&comma;174&comma;984&comma;52&comma;48&comma;201&comma;126&comma;176&comma;216&comma;190&comma;198&comma;148&comma;1955&comma;84&comma;61&comma;212&comma;75&comma;226&comma;211&comma;19&comma;99&comma;53&comma;2356&comma;106&comma;123&comma;69&comma;60&comma;28&comma;185&comma;88&comma;24&comma;108&comma;1327&comma;177&comma;15&comma;184&comma;22&comma;43&comma;125&comma;29&comma;217&comma;131&comma;378&comma;104&comma;191&comma;56&comma;163&comma;243&comma;59&comma;90&comma;244&comma;138&comma;1169&comma;135&comma;239&comma;236&comma;249&comma;145&comma;156&comma;197&comma;207&comma;266&comma;24110&comma;193&comma;188&comma;82&comma;17&comma;133&comma;210&comma;157&comma;105&comma;240&comma;1213&comma;255&comma;21&comma;153&comma;65&comma;209&comma;44&comma;258&comma;107&comma;72&comma;17314&comma;152&comma;221&comma;18&comma;63&comma;141&comma;225&comma;162&comma;46&comma;41&comma;5716&comma;262&comma;33&comma;39&comma;222&comma;218&comma;238&comma;223&comma;26&comma;77&comma;3620&comma;261&comma;180&comma;263&comma;118&comma;250&comma;172&comma;220&comma;27&comma;202&comma;4723&comma;154&comma;58&comma;143&comma;208&comma;182&comma;224&comma;256&comma;92&comma;252&comma;18730&comma;119&comma;113&comma;253&comma;124&comma;54&comma;70&comma;97&comma;245&comma;169&comma;7431&comma;149&comma;45&comma;51&comma;35&comma;100&comma;87&comma;242&comma;83&comma;170&comma;16732&comma;109&comma;248&comma;192&comma;228&comma;166&comma;178&comma;246&comma;229&comma;194&comma;4240&comma;130&comma;168&comma;64&comma;179&comma;159&comma;134&comma;200&comma;146&comma;230&comma;16549&comma;264&comma;234&comma;94&comma;78&comma;175&comma;139&comma;219&comma;155&comma;102&comma;21555&comma;144&comma;110&comma;231&comma;251&comma;91&comma;199&comma;115&comma;189&comma;183&comma;17162&comma;247&comma;112&comma;71&comma;73&comma;259&comma;257&comma;114&comma;101&comma;68&comma;23266&comma;121&comma;213&comma;136&comma;160&comma;140&comma;122&comma;196&comma;120&comma;111&comma;16193&comma;203&comma;233&comma;227&comma;103&comma;117&comma;206&comma;205&comma;137&comma;204&comma;128

(31)

GroupOrderK

11

(32)

CFrobeniusComplementG

C1&comma;218&comma;232&comma;194&comma;2492&comma;100&comma;186&comma;97&comma;2203&comma;169&comma;172&comma;25&comma;454&comma;49&comma;46&comma;44&comma;155&comma;40&comma;66&comma;157&comma;1856&comma;212&comma;179&comma;111&comma;1057&comma;52&comma;175&comma;14&comma;1738&comma;243&comma;244&comma;104&comma;1169&comma;76&comma;222&comma;259&comma;17810&comma;60&comma;84&comma;159&comma;13611&comma;253&comma;202&comma;214&comma;3112&comma;132&comma;53&comma;130&comma;14013&comma;43&comma;126&comma;102&comma;4116&comma;114&comma;166&comma;156&comma;8517&comma;123&comma;61&comma;165&comma;12218&comma;153&comma;131&comma;48&comma;21519&comma;168&comma;161&comma;188&comma;6920&comma;260&comma;87&comma;174&comma;7421&comma;22&comma;195&comma;139&comma;14123&comma;252&comma;187&comma;208&comma;25624&comma;235&comma;134&comma;120&comma;8226&comma;71&comma;246&comma;145&comma;14227&comma;127&comma;83&comma;181&comma;5428&comma;211&comma;146&comma;196&comma;24029&comma;201&comma;78&comma;225&comma;25830&comma;250&comma;150&comma;167&comma;6732&comma;236&comma;164&comma;33&comma;25734&comma;238&comma;73&comma;109&comma;20735&comma;98&comma;124&comma;263&comma;15836&comma;112&comma;248&comma;239&comma;5037&comma;190&comma;234&comma;162&comma;25538&comma;223&comma;68&comma;228&comma;13539&comma;62&comma;192&comma;197&comma;8942&comma;266&comma;79&comma;77&comma;10147&comma;265&comma;51&comma;96&comma;11955&comma;110&comma;144&comma;115&comma;25156&comma;138&comma;59&comma;191&comma;16357&comma;209&comma;217&comma;198&comma;21958&comma;92&comma;182&comma;154&comma;14363&comma;107&comma;177&comma;216&comma;15564&comma;160&comma;133&comma;88&comma;22665&comma;184&comma;176&comma;94&comma;15270&comma;180&comma;80&comma;149&comma;23772&comma;125&comma;148&comma;264&comma;22175&comma;230&comma;121&comma;193&comma;10881&comma;113&comma;118&comma;151&comma;24286&comma;170&comma;95&comma;245&comma;26199&comma;200&comma;213&comma;210&comma;106103&comma;204&comma;203&comma;117&comma;227128&comma;206&comma;137&comma;205&comma;233171&comma;189&comma;183&comma;231&comma;199229&comma;241&comma;254&comma;262&comma;247&comma;1&comma;652&comma;1504&comma;735&comma;846&comma;1327&comma;268&comma;1449&comma;14110&comma;15711&comma;9812&comma;10513&comma;8914&comma;14515&comma;23816&comma;21717&comma;8218&comma;23919&comma;22620&comma;26121&comma;7622&comma;22223&comma;20624&comma;12327&comma;11829&comma;26230&comma;9731&comma;3532&comma;26433&comma;12534&comma;4436&comma;13137&comma;7738&comma;10739&comma;4340&comma;15941&comma;19742&comma;23446&comma;20747&comma;18048&comma;11249&comma;10950&comma;15351&comma;14952&comma;7153&comma;21254&comma;11355&comma;10456&comma;18357&comma;15658&comma;10359&comma;19960&comma;18561&comma;23562&comma;12663&comma;13564&comma;16866&comma;13667&comma;18668&comma;21669&comma;8870&comma;11972&comma;16474&comma;24575&comma;9978&comma;22979&comma;25580&comma;26581&comma;18183&comma;24285&comma;20986&comma;26087&comma;17090&comma;9192&comma;20493&comma;22494&comma;19495&comma;17496&comma;237100&comma;167101&comma;190102&comma;192106&comma;108110&comma;116111&comma;140114&comma;198115&comma;243117&comma;154120&comma;122121&comma;213124&comma;253127&comma;151128&comma;256129&comma;147130&comma;179133&comma;188134&comma;165137&comma;252138&comma;231139&comma;178142&comma;173143&comma;227148&comma;257152&comma;249155&comma;228158&comma;214160&comma;161162&comma;266163&comma;189166&comma;219171&comma;191175&comma;246176&comma;232177&comma;223182&comma;203184&comma;218187&comma;205193&comma;210195&comma;259200&comma;230201&comma;247202&comma;263208&comma;233215&comma;248220&comma;250221&comma;236225&comma;241244&comma;251254&comma;258

(33)

GroupOrderC

10

(34)

IsMalnormalC&comma;G

true

(35)

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

GDihedralGroup7&colon;

CFrobeniusComplementG

C2&comma;73&comma;64&comma;5

(36)

IsMalnormalC&comma;G

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.)

GMathieuGroup10

GM10

(38)

SStabiliser1&comma;G

S < a permutation group on 10 letters with 7 generators >

(39)

GroupOrderS

72

(40)

This point stabiliser is a Frobenius group.

IsFrobeniusGroupS

true

(41)

Moreover, the action is Frobenius.

IsFrobeniusPermGroupS

true

(42)

The Frobenius complement in S is a quaternion group.

AreIsomorphicFrobeniusComplementS&comma;QuaternionGroup

true

(43)

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.

See Also

GroupTheory

GroupTheory[IsNilpotent]