Frobenius group

In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial elementfixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius. In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial elementfixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius. A 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 (1901); there is still no proof of this theorem that does not use character theory, although see .) 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 ithas 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. 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: