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 th eory. Some problems, such as the computation of the number of isomorphism c lasses of Frobenius groups subject to various restrictions on orders, are f urther reduced to elementary number theory. The starting point is the bijec tion between the set of isomorphism classes of Frobenius groups with commut ative 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 c enter Z is an abelian extension of Q. All Frobenius complements and their a ssociated rings are explicitly computed here in terms of simple numerical i nvariants. The finite modules of such a ring are sums of indecomposable one s, and the indecomposable ones are shown to correspond to powers of unramif ied (over Q) maximal ideals of the ring of integers of Z which do not conta in the order of the Frobenius complement.