Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field K of positive characteristic p, necessarily a prime number. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory. Within finite group theory, character-theoretic results proved by Richard Brauer using modular representation theory played an important role in early progress towards the classification of finite simple groups, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because their Sylow 2-subgroups were too small in an appropriate sense. 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. If the characteristic p of K does not divide the order |G|, then modular representations are completely reducible, as with ordinary (characteristic 0) representations, by virtue of Maschke's theorem. In the other case, when |G| ≡ 0 mod p, the process of averaging over the group needed to prove Maschke's theorem breaks down, and representations need not be completely reducible. 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. History The earliest work on representation theory over finite fields is by who showed that when p does not divide the order of the group, the representation theory is similar to that in characteristic 0. He also investigated modular invariants of some finite groups. The systematic study of modular representations, when the characteristic p divides the order of the group, was started by and was continued by him for the next few decades. Example Finding a representation of the cyclic group of two elements over F2 is equivalent to the problem of finding matrices whose square is the identity matrix. 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