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-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.
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.
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.
However, this is not a Frobenius action; to get a Frobenius permutation group, use FrobeniusPermRep.
Now we can compute the Frobenius kernel and complement, and determine their orders.
C≔ < a permutation group on 11 letters with 6 generators >
Of course, we obtain the same result by computing the Frobenius kernel and complement of G itself.
The Frobenius complement in a Frobenius dihedral group is a subgroup of order two.
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.)
S≔ < a permutation group on 10 letters with 7 generators >
This point stabiliser 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.
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