Frobenius group

In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius.

Structure

The subgroup H of a Frobenius group G fixing a point of the set X is called the Frobenius complement. The identity element together with all elements not in any conjugate of H form a normal subgroup called the Frobenius kernel K. (This is a theorem due to Frobenius.) The Frobenius group G is the semidirect product of K and H:

G = K ⋊ H.

Both the Frobenius kernel and the Frobenius complement have very restricted structures. J. G. Thompson proved that the Frobenius kernel K is a nilpotent group. If H has even order then K is abelian. The Frobenius complement H has the property that every subgroup whose order is the product of 2 primes is cyclic; this implies that its Sylow subgroups are cyclic or generalized quaternion groups. Any group such that all Sylow subgroups are cyclic is metacyclic: this means it is the extension of two cyclic groups. If a Frobenius complement H is not solvable then Zassenhaus showed that it has a normal subgroup of index 1 or 2 that is the product of SL₂(5) and a metacyclic group of order coprime to 30. If a Frobenius complement H is solvable then it has a normal metacyclic subgroup such that the quotient is a subgroup of the symmetric group on 4 points.

The Frobenius kernel K is uniquely determined by G as it is the Fitting subgroup, and the Frobenius complement is uniquely determined up to conjugacy by the Schur-Zassenhaus theorem. In particular a finite group G is a Frobenius group in at most one way.

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 $x\mapsto ax+b$, $a\neq 0$ with its natural action on F_q is a Frobenius group. The preceding example corresponds to the case F₃, the field with three elements.

• More generally, the group of upper 2 × 2 invertible triangular matrices of determinant 1 over any finite field of order at least 3 is a Frobenius group. The Frobenius kernel is the subgroup of strictly upper triangular matrices (with diagonal elements equal to 1), and the complement is the subgroup of diagonal matrices.

• 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.

• 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.

Representation theory

The irreducible complex representations of a Frobenius group G can be read off from those of H and K. There are two types of irreducible representations of G:

• Any irreducible representation R of H gives an irreducible representation of G using the quotient map from G to H (that is, as a restricted representation). These give the irreducible representations of G with K in their kernel.
• If S is any non-trivial irreducible representation of K, then the corresponding induced representation of G is also irreducible. These give the irreducible representations of G with K not in their kernel.

References

• D. S. Passman, Permutation groups, Benjamin 1968