Suppose G is a Frobenius group consisting of permutations of a set X. A
subgroupH of G fixing a point of X is called a Frobenius complement. The identity element together with all elements not in any conjugate of H form a
normal subgroup called the Frobenius kernelK. (This is a theorem due to
Frobenius (1901); there is still no proof of this theorem that does not use
character theory, although see [1].) The Frobenius group G is the
semidirect product of K and H:
.
Both the Frobenius kernel and the Frobenius complement have very restricted structures.
J. G. Thompson (
1960) 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 called a
Z-group, and in particular must be a
metacyclic group: 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 SL(2,5) and a metacyclic group of order coprime to 30. In particular, if a Frobenius complement coincides with its derived subgroup, then it is isomorphic with SL(2,5). 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. A finite group is a Frobenius complement if and only if it has a faithful, finite-dimensional representation over a finite field in which non-identity group elements correspond to linear transformations without nonzero fixed 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 fieldFq 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 productK.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.
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. 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.
Alternative definitions
There are a number of group theoretical properties which are interesting on their own right, but which happen to be equivalent to the group possessing a permutation representation that makes it a Frobenius group.
G is a Frobenius group if and only if G has a proper, nonidentity subgroup H such that H ∩ Hg is the identity subgroup for every g ∈ G − H, i.e.H is a
malnormal subgroup of G.
This definition is then generalized to the study of trivial intersection sets which allowed the results on Frobenius groups used in the classification of
CA groups to be extended to the results on
CN groups and finally the
odd order theorem.
Assuming that is the
semidirect product of the normal subgroup K and complement H, then the following restrictions on
centralizers are equivalent to G being a Frobenius group with Frobenius complement H:
The
centralizer CG(k) is a subgroup of K for every nonidentity k in K.