The first part of the book centers around the isomorphism problem for finite groups; i.e. which properties of the finite group G can be determined by the integral group ring ZZG ? The authors have tried to present the results more or less selfcontained and in as much generality as possible concerning the ring of coefficients. In the first section, the class sum correspondence and some related results are derived. This part is the proof of the subgroup rigidity theorem (Scott - Roggenkamp; Weiss) which says that a finite subgroup of the p-adic integral group ring of a finite p-group is conjugate to a subgroup of the finite group. A counterexample to the conjecture of Zassenhaus that group basis are rationally conjugate, is presented in the semilocal situation (Scott - Roggenkamp). To this end, an extended version of Clifford theory for p-adic integral group rings is presented. Moreover, several examples are given to demonstrate the complexity of the isomorphism problem. The second part of the book is concerned with various aspects of the structure of rings of integers as Galois modules. It begins with a brief overview of major results in the area; thereafter the majority of the text focuses on the use of the theory of Hopf algebras. It begins with a thorough and detailed treatment of the required foundational material and concludes with new and interesting applications to cyclotomic theory and to elliptic curves with complex multiplication. Examples are used throughout both for motivation, and also to illustrate new ideas.
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.
Series: Oberwolfach Seminars : Book 18
Number Of Pages: 210
Publisher: SPRINGER VERLAG GMBH
Country of Publication: CH
Dimensions (cm): 24.41 x 16.99
Weight (kg): 0.36