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Brauer's first main theorem states that the number of blocks of a finite group that have a given ''p''-subgroup as defect group is the same as the corresponding number for the normalizer in the group of that ''p''-subgroup.
The easiest block structure to analyse with non-trivial defect group is when the latter is cyclic. Then there are only finitely many isomorphism types of indecomposable modules in the block, and the structure of the block is by now well understood, by virtue of work of Brauer, [[Everett C. Dade|E.C. Dade]], J.A. Green and [[John Griggs Thompson|J.G. Thompson]], among others. In all other cases, there are infinitely many isomorphism types of indecomposable modules in the block.
Blocks whose defect groups are not cyclic can be divided into two types: tame and wild. The tame blocks (which only occur for the prime 2) have as a defect group a [[dihedral group]], [[semidihedral group]] or (generalized) [[quaternion group]], and their structure has been broadly determined in a series of papers by [[Karin Erdmann]]. The indecomposable modules in wild blocks are extremely difficult to classify, even in principle.
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