The smallest Frobenius group is the symmetric group of degree .
A different permutation group isomorphic to the symmetric group of degree is a Frobenius group, but is not Frobenius as a permutation group.
The dihedral group is Frobenius if, and only, if, is odd.
We construct here a Frobenius subgroup of order 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.
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 has a point stabilizer of order . (This is sometimes referred to as a Mathieu group of degree .)
This point stabilizer is a Frobenius group.
Moreover, the action is Frobenius.
The Frobenius complement in is a quaternion group.