The structure of blocks with a Klein four defect group. (Q543334)

Puig conjectured that for a given \(p\)-group \(P\) there are only finitely many isomorphism classes of interior \(P\)-algebras arising as source algebras of \(p\)-blocks of finite groups. In the present paper the case of \(P\) being a Klein four group is shown. In order to do this the authors show in fact that no simple module of a block with Klein four defect group \(P\) can have a source which is an endo-permutation module with image of infinite order in the Dade group. The proof then reduces to the quasi-simple case and proceeds then by a case-by-case analysis. All along the proof it is taken care to stay as general as possible in view of a possible generalisation to the odd \(p\) case.