G-theory of group rings for finite groups

Abstract

In this thesis we investigate Quillen's G-theory of group rings mostly focusing on the case of finite groups. We study the Hambleton-Taylor-Williams decomposition conjecture for G-theory of the integral group rings. The conjecture expresses n-th G-group of the integral group ring as a direct sum of G-groups of maximal orders in the simple components of QG with certain integers inverted. The HTW-conjecture generalizes the results of Lenstra and Webb on abelian groups. Webb and Yao found a counterexample to the HTW-decomposition in degree 1 but nevertheless they still expected the conjecture to hold for solvable groups. Using the results of Keating we show that the solvable group SL(2, F3) is a counterexample to the conjectured decomposition. Using the methods from modular representation theory we prove useful inequality for ranks of G-groups in degree 1. It is also shown that the HTW-decomposition gives a correct prediction for the torsion subgroup in degree 1 for all finite groups G. Furthermore, we prove that the ranks of G-groups in degree n agree with the prediction of the conjecture in all degrees apart from the degree n = 1.