Frobenius group

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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.

Contents

1 Structure
2 Examples
3
4 References

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\ne 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.

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

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