Frobenius groups and classical maximal orders

The analysis of the set of isomorphism classes of Frobenius groups with com mutative Frobenius kernel is reduced here to "abelian" algebraic number theory. Some problems, such as the computation of the number of isomorphism classes of Frobenius groups subject to various restrictions on orders, are further reduced to elementary number theory. The starting point is the bijection between the set of isomorphism classes of Frobenius groups with commutative Frobenius kernel and with given Frobenius complement G and the set of G-semi-linear isomorphism classes of finite modules over a ring naturally associated with G, This ring is a maximal order in a simple algebra whose center Z is an abelian extension of Q. All Frobenius complements and their associated rings are explicitly computed here in terms of simple numerical invariants. The finite modules of such a ring are sums of indecomposable ones, and the indecomposable ones are shown to correspond to powers of unramified (over Q) maximal ideals of the ring of integers of Z which do not contain the order of the Frobenius complement.