On the sources of simple modules in nilpotent blocks

Abstract

Let G be a finite group and let k be an algebraically closed field of characteristic p. If b is a nilpotent block of kG with defect group P, then there is a unique isomorphism class of simple kGb-modules and Puig proved that the source of this module is an endo-permutation kP-module. It is conjectured that the image of this source is always torsion in the Dade group. Let H be a finite group and let P be a p-subgroup of Aut ( H ) . Also let c be a defect zero block of kH. If c is P-stable and Br P ( c ) ≠ 0 , then c is a nilpotent block of k ( H ⋊ P ) and k ( H ⋊ P ) c has P as a defect group. In this paper, we will investigate the sources of the simple k ( H ⋊ P ) c -modules when P ≅ C p × C p . Suppose that we can find an H and c as above such that a source of a simple k ( H ⋊ P ) c -module is not torsion in the Dade group. Then we can find H and c as above with H a central p ′ -extension of a simple group. When p ⩾ 3 we show that H can be found in a quite restrictive subset of simple groups.