Rationality of blocks of quasi-simple finite groups

• Main Author: Farrell, N.
• Published: City, University of London 2017
• Subjects: 512; QA Mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.716133

The Morita Frobenius number of an algebra is the number of Morita equivalence classes of its Frobenius twists. Introduced by Kessar in 2004, these numbers are important in the context of Donovan's conjecture for blocks of finite group algebras. Let P be a finite ℓ-group. Donovan's conjecture states that there are finitely many Morita equivalence classes of blocks of finite group algebras with defect groups isomorphic to P. Kessar proved that Donovan's conjecture holds if and only if Weak Donovan's conjecture and the Rationality conjecture hold. Our thesis relates to the Rationality conjecture, which states that there exists a bound on the Morita Frobenius numbers of blocks of finite group algebras with defect groups isomorphic to P, which depends only on SPS. In this thesis we calculate the Morita Frobenius numbers, or produce a bound for the Morita Frobenius numbers, of many of the blocks of quasi-simple finite groups. We also discuss the issues faced in the outstanding blocks and outline some possible approaches to solving these cases.