determine whether a group is a Frobenius permutation group
determine whether a group is a Frobenius group
compute the Frobenius kernel of a Frobenius group
compute the Frobenius complement of a Frobenius group
compute a Frobenius permutation group isomorphic to a given Frobenius group
IsFrobeniusPermGroup( G )
IsFrobeniusGroup( G )
FrobeniusKernel( G )
FrobeniusComplement( G )
FrobeniusPermRep( G )
a permutation group
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.
The smallest Frobenius group is the symmetric group of degree 3.
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 ≔ Group⁡Perm⁡1,2,3,6,4,5,Perm⁡1,3,4,2,5,6
H ≔ FrobeniusPermRep⁡G
The dihedral group Dn is Frobenius if, and only, if, n is odd.
We construct here a Frobenius subgroup of order 110 in the first Janko group.
a,b ≔ op⁡Generators⁡JankoGroup⁡1:
u ≔ `.`⁡`.`⁡a,b−2,b,a,b:PermOrder⁡u
v ≔ `.`⁡`.`⁡a,b,b−2,`.`⁡a,b,a,b,a,b,b,a,b,a,b,b3,`.`⁡a,b,b2:PermOrder⁡v
G ≔ Group⁡u,v:GroupOrder⁡G
However, this is not a Frobenius action; to get a Frobenius permutation group, use FrobeniusPermRep.
P ≔ FrobeniusPermRep⁡G
Now we can compute the Frobenius kernel and complement, and determine their orders.
K ≔ FrobeniusKernel⁡P
C ≔ FrobeniusComplement⁡P
Of course, we obtain the same result by computing the Frobenius kernel and complement of G itself.
K ≔ FrobeniusKernel⁡G
C ≔ FrobeniusComplement⁡G
The Frobenius complement in a Frobenius dihedral group is a subgroup of order two.
G ≔ DihedralGroup⁡7:
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 ≔ MathieuGroup⁡10
S ≔ Stabilizer⁡1,G
This point stabilizer is a Frobenius group.
Moreover, the action is Frobenius.
The Frobenius complement in S is a quaternion group.
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.
Download Help Document
What kind of issue would you like to report? (Optional)