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 (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. 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. CH(k) = 1 for every nonidentity k in K. CG(h) ≤ H for every nonidentity h in H. References B. Huppert, Endliche Gruppen I, Springer 1967 I. M. Isaacs, Character theory of finite groups, AMS Chelsea 1976 D. S. Passman, Permutation groups'', Benjamin 1968 Permutation groups Finite groups