Frobenius Group - Examples

Examples

• The smallest example is the symmetric group on 3 points, with 6 elements. The Frobenius kernel K has order 3, and the complement H has order 2.

• For every finite field F_q with q (> 2) elements, the group of invertible affine transformations, acting naturally on F_q 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 in the prime field F₂ (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 field F of order q and some element z of this field of order p. The Frobenius complement H is the cyclic subgroup generated by the diagonal matrix whose i,i'th entry is zi. The Frobenius kernel K is the Sylow q-subgroup of GL(d,q) consisting of upper triangular matrices with ones on the diagonal. The kernel K has nilpotency class d −1, and the semidirect product KH is a Frobenius group.