Summary: A Construction of Endopermutation Modules
J.L. Alperin \Lambda
Endopermutation modules for finite pgroups appear in the representation theory of finite
groups in several ways, for example as sources of simple modules and in connection with
equivalences between blocks [6, 7, 11]. The main problem is their complete description and
classification. We shall give a simple new construction of such modules, study their properties
and apply the results to the group of endotrivial modules, determining its rank.
Let P be a fixed finite pgroup and k an algebraically closed field of characteristic p. All
modules for kP will be assumed finitedimensional and all P sets will be finite. If X is a
P set let \Delta(X ) be the kernel of the map from the kPmodule kX to the trivial modules
k sending each element of X to 1 (the augmentation map). Recall that a kPmodule U
is an endopermutation module if the kPmodule Hom k (U; U) = End k (U) is a permutation
module and U is an endotrivial module if Hom k (U; U) is the direct sum of k and a free
Theorem 1. The module \Delta(X ) is an endopermutation module.
Theorem 2. The module \Delta(X ) is indecomposable if, and only if, no orbit of P on X is a
homomorphic image, as P set, of another orbit of P on X.
Theorem 3. If \Delta(X ) is indecomposable and Q is a subgroup of P then the class [Br P
Q (\Delta(X ))]