Endo-permutation Modules, a Guided Tour

This survey paper gives an overview of the recent classification of all endopermutation kP -modules, where P is a finite p-group and k is a field of characteristic p. It is an expanded version of two talks given in April 2005 during a special workshop on endo-permutation modules, organized within the program of the Bernoulli Centre of EPFL. The classification of endo-permutation modules was completed in 2004, a quarter of a century after the first decisive results of Dade [Da2] in 1978. The final results are due to the combined efforts of several authors during the years 1998– 2004. The first crucial step was the classification of all modules in an important subclass, namely the class of endo-trivial kP -modules. This appears in the work of Carlson and Thévenaz [CaTh1], [CaTh2], [CaTh3]. The classification of all endo-permutation kP -modules when P is extraspecial (or almost extraspecial), due to Bouc and Mazza [BoMa], was obtained shortly afterwards. The final completion of the classification in all cases is due to Bouc [Bo7], based on the above mentioned papers and on several aspects of his previous work [Bo2], [Bo4], [Bo5]. The important role of relative syzygies had been discovered a few years before by Alperin [Al3]. This survey will not follow the chronological order of the various publications, but rather what appears to be a logical development of the subject, at least in our opinion. I am grateful to Serge Bouc, Jon Carlson, and Nadia Mazza for helpful comments on a preliminary version of this paper.