Abelian Frobenius kernels and modules over number rings Original Research Article

Some categories of groups (typically involving groups, possibly infinite, with abelian Frobenius kernels) are shown to be equivalent to categories whose objects are modules over groups and rings. The rings in question are associated with algebraic number fields, and the category equivalences allow some applications of number theory to group theory. In particular the number of isomorphism classes of metabelian Frobenius groups with kernel of order n and complement of order m is calculated in terms of the prime factorization of n and the structure of the Euler group image.