3, and the complement H has order 2. For every finite field Fq with q (> 2) elements, the group of invertible affine transformations , acting naturally on Fq is a Frobenius group. The preceding example corresponds to the case F3, the field with three elements. Another example is provided by the subgroup of order 21 of the collineation group of the Fano plane generated by a 3-fold symmetry σ fixing a point and a cyclic permutation τ of all 7 points, satisfying στ = τ2σ. Identifying F8× with the Fano plane, σ can be taken to be the restriction of the Frobenius automorphism σ(x) = x2 of F8 and τ to be multiplication by any element not 0 or 1 (i.e. a generator of the cyclic multiplicative group of F8). This Frobenius group acts simply transitively on the 21 flags in the Fano plane, i.e. lines with marked points. The dihedral group of order 2n with n odd is a Frobenius group with complement of order 2. More generally if K is any abelian group of odd order and H has order 2 and acts on K by inversion, then the semidirect product K.H is a Frobenius group. Many further examples can be generated by the following constructions. If we replace the Frobenius complement of a Frobenius group by a non-trivial subgroup we get another Frobenius group. If we have two Frobenius groups K1.H and K2.H then (K1 × K2).H is also a Frobenius group. If K is the non-abelian group of order 73 with exponent 7, and H is the cyclic group of order 3, then there is a Frobenius group G that is an extension K.H of H by K. This gives an example of a Frobenius group with non-abelian kernel. This was the first example of Frobenius group with nonabelian kernel (it was constructed by Otto Schmidt). If H is the group SL2(F5) of order 120, it acts fixed point freely on a 2-dimensional vector space K over the field with 11 elements. The extension K.H is the smallest example of a non-solvable Frobenius group. The subgroup of a Zassenhaus group fixing a point is a Frobenius group. Frobenius groups whose Fitting subgroup has arbitrarily large nilpotency class were constructed by Ito: Let q be a prime power, d a positive integer, and p a prime divisor of q −1 with d ≤ p. Fix some