Nadia Mazza

Endotrivial modules for finite groups of Lie type.

Abstract 1. Introduction Let G be a finite group and k be a field of characteristic p > 0. An endotrivial kG-module is a finitely generated kG-module M whose k-endomorphism ring is isomorphic to a trivial module in the stable module category. That is, M is an endotrivial module provided where P is a projective kG-module. Now recall that as kG-modules, where M * = Hom k (M, k) is the k-dual of M. Hence, the functor “ ” induces an equivalence on the stable module category and the collection of all endotrivial modules makes up a part of the Picard group of all stable equivalences of kG-modules. In particular, equivalence classes of endotrivial modules modulo projective summands form a group that is an essential part of the group of stable self-equivalences.

ENDOTRIVIAL MODULES FOR THE SYMMETRIC AND ALTERNATING GROUPS

In this paper we determine the group of endotrivial modules for certain symmetric and alternating groups in characteristic

Endotrivial modules for

p

p

. If

-solvable groups.

p=2

The Dade group of a fusion system.

, then the group is generated by the class of

On Oliver’s p-group conjecture: II

\Omega^n(k)

Endotrivial modules for the sporadic simple groups and their covers

except in a few low degrees. If

Glauberman's and Thompson's theorems for fusion systems

p >2

THE GROUP OF ENDOTRIVIAL MODULES FOR THE SYMMETRIC AND ALTERNATING GROUPS

, then the group is only determined for degrees less than