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Blocks with basic algebras of dimension at most 12
In [Li18b] Markus Linckelmann calculated the [math]k[/math]-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9 where no block with that basic algebra is identified. See Blocks with basic algebras of low dimension for a description of these results.
Morita invariance of the isomorphism type of a defect group
It is not known whether there exist Morita equivalent blocks with non-isomorphic defect groups. In general this is a difficult problem, subsuming the modular isomorphism problem for [math]p[/math]-groups. Basic Morita equivalences do preserve the isomorphism type of a defect group, and part of the difficulty in resolving the question is the lack of examples of Morita equivalent blocks which are not also basic Morita equivalent (this is not to say that every known Morita equivalence is basic). A survey may be found in [NS18].