Cohomology algebras of 2-blocks of finite groups with defect groups of rank two.
Let \(G\) be a finite group, \(k\) an algebraically closed field of characteristic \(p\) dividing the order of \(G\). Let \(B\) be a block algebra of \(kG\) with defect group \(D\), and let \((D,b_D)\) be a maximal \(B\)-subpair. By definition, Linckelmann's cohomology algebra \(H^*(G,B)\) of \(B\) consists of elements \(\sigma\in H^*(G,k)\) satisfying \(\text{res}_P\sigma={^g\text{res}_P\sigma}\) for all \(P\leq D\) and all \(g\in N_G(D,b_P)\). In this paper the authors calculate the cohomology algebra of \(2\)-blocks with dihedral, semidihedral, quaternion, and wreathed \(2\)-groups. In these cases, \(H^*(G,B)\) is determined by particular \(B\)-subpairs which are specified. The relationship between the cohomology algebras of the Brauer corresponding blocks is also investigated. If \(C\) is the Brauer correspondent of \(B\) in a subgroup \(H\) containing \(N_G(D)\), then under certain conditions, \(H^*(G,B)\) is a subgroup of \(H^*(H,C)\). A transfer map from \(H^*(H,C)\) to \(H^*(G,B)\) is given explicitly.
scientific article; zbMATH DE number 5115341
scientific article