Blocks and indecomposable modules over twisted group algebras (Q1057358)

Let G be a finite group, let F be an arbitrary field of characteristic \(p>0\), and let \(\alpha \in Z^ 2(G,F^*)\). Denote by \(F^{\alpha}G\) the corresponding twisted group algebra. The aim of this paper is to establish the following generalization of a theorem of \textit{J. A. Green} [Math. Z. 79, 100-115 (1962; Zbl 0233.20006)]. Theorem. Let B be a block of \(F^{\alpha}G\) with defect group D and let \(\Delta\) : \(g\to (g,g)\) be the diagonal map of G into \(G\times G\). Define \(\alpha^{-1}\times \alpha: (G\times G)\times (G\times G)\to F^*\) by \((\alpha^{-1}\times \alpha)((g_ 1,g_ 2),(g'\!_ 1,g'\!_ 2))=\alpha^{-1}(g_ 1,g'\!_ 1)\alpha (g_ 2,g'\!_ 2).\) Then the indecomposable \(F^{\alpha^{-1}\times \alpha}(G\times G)\)-module B has vertex \(\Delta\) (D). Furthermore, D is an intersection of two Sylow p-subgroups of G.