BRAUER INDECOMPOSABILITY OF SCOTT MODULES WITH SEMIDIHEDRAL VERTEX

We present a sufficient condition for the kG-Scott module with vertex P to remain indecomposable under the Brauer construction for any subgroup Q of P as k[QC(G)(Q)]-module, where k is a field of characteristic 2, and P is a semidihedral 2-subgroup of a finite group G. This generalizes results for the cases where P is abelian or dihedral. The Brauer indecomposability is defined by R. Kessar, N. Kunugi and N. Mitsuhashi. The motivation of this paper is the fact that the Brauer indecomposability of a p-permutation bimodule (where p is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method due to Broue, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence.