Also, a general result on embedding of elements of order 2 in finite groups called the Z* theorem, proved by George Glauberman using the theory developed by Brauer, was particularly useful in the classification program.
Much of the discussion below implicitly assumes that the field K is sufficiently large (for example, K algebraically closed suffices), otherwise some statements need refinement.
The systematic study of modular representations, when the characteristic p divides the order of the group, was started by Brauer (1935) and was continued by him for the next few decades.
Over every field of characteristic other than 2, there is always a basis such that the matrix can be written as a diagonal matrix with only 1 or −1 occurring on the diagonal, such as Over F2, there are many other possible matrices, such as Over an algebraically closed field of positive characteristic, the representation theory of a finite cyclic group is fully explained by the theory of the Jordan normal form.
When the order of G is divisible by the characteristic of K, the group algebra is not semisimple, hence has non-zero Jacobson radical.
Such results can be applied in group theory to problems not directly phrased in terms of representations.
When K is algebraically closed of positive characteristic p, there is a bijection between roots of unity in K and complex roots of unity of order coprime to p. Once a choice of such a bijection is fixed, the Brauer character of a representation assigns to each group element of order coprime to p the sum of complex roots of unity corresponding to the eigenvalues (including multiplicities) of that element in the given representation.
The Brauer character of a representation determines its composition factors but not, in general, its equivalence type.
These are integral (though not necessarily non-negative) combinations of the restrictions to elements of order coprime to p of the ordinary irreducible characters.
When the field F has characteristic 0, or characteristic coprime to the group order, there is still such a decomposition of the group algebra F[G] as a sum of blocks (one for each isomorphism type of simple module), but the situation is relatively transparent when F is sufficiently large: each block is a full matrix algebra over F, the endomorphism ring of the vector space underlying the associated simple module.
To obtain the blocks, the identity element of the group G is decomposed as a sum of primitive idempotents in Z(R[G]), the center of the group algebra over the maximal order R of F. The block corresponding to the primitive idempotent e is the two-sided ideal e R[G].
Using the ring R as above, with residue field K, the identity element of G may be decomposed as a sum of mutually orthogonal primitive idempotents (not necessarily central) of K[G].
When a projective module is lifted, the associated character vanishes on all elements of order divisible by p, and (with consistent choice of roots of unity), agrees with the Brauer character of the original characteristic p module on p-regular elements.
This is referred to as the decomposition matrix, and is frequently labelled D. It is customary to place the trivial ordinary and Brauer characters in the first row and column respectively.
The product of the transpose of D with D itself results in the Cartan matrix, usually denoted C; this is a symmetric matrix such that the entries in its j-th row are the multiplicities of the respective simple modules as composition factors of the j-th projective indecomposable module.
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.
The order of the defect group of a block has many arithmetical characterizations related to representation theory.