A derived equivalence for blocks with dihedral defect groups (Q1322526)

Let \(G\) be a finite group, \(k\) be an algebraically closed field of positive characteristic \(p\). The term ``block'' means a primitive idempotent of the center of \(kG\). If \(b\) is a block, the author calls the group \(N_ G(P_ \gamma)/PC_ G(P)\) internal quotient of \(b\), where \(P_ \gamma\) is a maximal local pointed group contained in \(G\) in Puig's terminology. Two blocks \(b_ 1\) and \(b_ 2\) of the groups \(G_ 1\) and \(G_ 2\) are derived equivalent if the algebras \(kG_ 1b_ 1\) and \(kG_ 2b_ 2\) are so. The main result of the paper is the following Theorem: Any two blocks with a common dihedral defect group having three isomorphism classes of simple modules are derived equivalent. In particular, for a defect group which is a Klein four group the author obtains the following Corollary: If \(b\) is a block of a finite group \(G\) having a Klein four group \(P\) as a defect group, then \(kGb\) and \(k(P \rtimes E)\) are derived equivalent, where \(P \rtimes E\) is the semidirect product of \(P\) by the internal quotient \(E\) of \(b\).