Groups of permutation projective dimension two

Abstract

In previous papers we developed a version of homological algebra for Z [ G ] Z[G] modules ( G G a finite group) using summands of permutation modules in place of projective modules. The resulting theory is more discriminating than the usual homological algebra. For example, an f.g. Z Z -torsion free Z [ G ] Z[G] module is either projective or has infinite projective dimension, whereas for G G cyclic, all f.g. Z [ G ] Z[G] modules have permutation projective dimension one. In this paper we apply results of Endo and Miyata on permutation projective modules to characterize groups of dimension two.