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We construct blocks of finite groups with arbitrarily large O -Morita Frobenius numbers. There are no known examples of two blocks defined over O , with isomorphic defect groups, that are not Morita equivalent but the corresponding blocks defined over k are. Therefore, the above strongly suggests that Morita Frobenius numbers are also unbounded, which would answer a question of Benson and Kessar. [ABSTRACT FROM AUTHOR]

We calculate examples of Picard groups for 2-blocks with abelian defect groups with respect to a complete discrete valuation ring. These include all blocks with abelian 2-groups of 2-rank at most three with the exception of the principal block of J 1. In particular this shows directly that all such Picard groups are finite and Picent, the group of Morita auto-equivalences fixing the centre, is trivial. These are amongst the first calculations of this kind. Further we prove some general results concerning Picard groups of blocks with normal defect groups as well as some other cases. [ABSTRACT FROM AUTHOR]