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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

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A permutation group  is a Frobenius group if it is transitive, has a non-trivial point stabilizer, and no non-trivial element of  fixes more than one point.

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The IsFrobeniusPermGroup( G ) command returns true if the permutation group G is a Frobenius group, and returns false otherwise.

• 

An abstract group  is a Frobenius group if it has a proper, non-trivial malnormal subgroup self-centralizing subgroup , called a Frobenius complement. In this case,  has a normal (even characteristic) subgroup , called the Frobenius kernel, consisting of the identity element of  and the elements of  that do not belong to any conjugate of  in .

• 

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  is a Frobenius permutation group, then  is Frobenius as an abstract group, with the stabilizer of a point being a Frobenius complement in . Conversely, if  is Frobenius as an abstract group, then the action of  on the cosets of a Frobenius complement is faithful and is Frobenius as a permutation group, and so  is isomorphic to the corresponding Frobenius permutation group,

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The Frobenius kernel of a Frobenius group  is uniquely defined, because a group can be a Frobenius group in at most one way. The Frobenius complement of a Frobenius group  is well-defined up to conjugacy in .

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If  is a Frobenius group, the FrobeniusKernel( G ) command returns the Frobenius kernel of .  If  is not Frobenius, an exception is raised.

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If  is a Frobenius group, the FrobeniusComplement( G ) command returns a Frobenius complement of .  If  is not Frobenius, an exception is raised.

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For a Frobenius group G, the FrobeniusPermRep( G ) command returns a Frobenius permutation group isomorphic to . It is permutation isomorphic to the action on  on the cosets of a Frobenius complement in .

Examples

The smallest Frobenius group is the symmetric group of degree .

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

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

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

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

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

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

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

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

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This point stabilizer is a Frobenius group.

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Moreover, the action is Frobenius.

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The Frobenius complement in  is a quaternion group.

(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.

See Also

GroupTheory

GroupTheory[IsNilpotent]

 


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