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'''Modular representation theory''' is a branch of [[mathematics]], and that part of [[representation theory]] that studies [[linear representation]]s of [[finite group|finite groups]]
naturally in other branches of mathematics, such as [[algebraic geometry]], [[coding theory]], [[combinatorics]] and [[number theory]].
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[[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 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 of ''K'' does not divide the order of the group, ''G'', then modular representations are completely reducible, as with ''ordinary''
(characteristic 0) representations, by virtue of [[Maschke's theorem]]. The proof of Maschke's theorem relies on being able to divide by the group order, which is not meaningful when the order of ''G'' is divisible by the characteristic of ''K''. In that case, representations need not be
completely reducible, unlike the ordinary (and the coprime characteristic) case. Much of the discussion below implicitly assumes
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