I Some general facts -- II Some notes on representation theory -- III The leading coefficient of units -- IV Class sum correspondence -- V More on the class sum correspondence -- VI Subgroup rigidity -- VII Global units -- VIII Locally isomorphic group rings -- IX Zassenhaus conjecture -- X Variations of the Zassenhaus conjecture -- XI Group Extensions -- XII Class sums of p-elements -- XIII Clifford theory revisited -- XIV Examples -- I Introduction and Review of the Tame Case -- II Hopf Orders -- III Principal Hornogeneous Spaces -- IV Arithmetic Applications:- The Cyclotomic Case -- V Arithmetic Applications:- The Elliptic Case -- References