Editing Modular representation theory (section)

== Defect groups ==

To each block ''B'' of the group algebra ''K''[''G''], Brauer associated a certain ''p''-subgroup, known as its '''defect group''' (where ''p'' is the characteristic of ''K''). Formally, it is the largest ''p''-subgroup
''D'' of ''G'' for which there is a [[Brauer's three main theorems|Brauer correspondent]] of ''B'' for the
subgroup <math>DC_G(D)</math>, where <math>C_G(D)</math> is the [[centralizer]] of ''D'' in ''G''.

The defect group of a block is unique up to conjugacy and has a strong influence on the structure of the block. For example, if the defect group is trivial, then the block contains just one simple module, just one ordinary character, the ordinary and Brauer irreducible characters agree on elements of order prime to the relevant characteristic ''p'', and the simple module is projective. At the other extreme, when ''K'' has characteristic ''p'', the [[Sylow]] ''p''-subgroup of the finite group ''G'' is a defect group for the principal block of ''K''[''G''].

The order of the defect group of a block has many arithmetical characterizations related to representation theory. It is the largest invariant factor of the Cartan matrix of the block, and occurs with
multiplicity one. Also, the power of ''p'' dividing the index of the defect group of a block is the [[greatest common divisor]] of the powers of ''p'' dividing the dimensions of the simple modules in that block, and this coincides with the greatest common divisor of the powers of ''p'' dividing the degrees of the ordinary irreducible characters in that block.

Other relationships between the defect group of a block and character theory include Brauer's result that if no conjugate of the ''p''-part of a group element ''g'' is in the defect group of a given block, then each irreducible character in that block vanishes at ''g''. This is one of many consequences of Brauer's second main theorem.

The defect group of a block also has several characterizations in the more module-theoretic approach to block theory, building on the work of [[Sandy Green (mathematician)|J. A. Green]], which associates a ''p''-subgroup
known as the '''vertex''' to an indecomposable module, defined in terms  of '''relative projectivity''' of the module. For example, the vertex of each indecomposable module in a block is contained (up to conjugacy)
in the defect group of the block, and no proper subgroup of the defect group has that property.

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.