Modular representation theory: Difference between revisions

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{{Short description| Studies linear representations of finite groups over a field K of positive characteristic p}}
'''Modular representation theory''' is a branch of [[mathematics]], and thatis the part of [[representation theory]] that studies [[linear representation]]s of [[finite group]]s over a [[field (mathematics)|field]] ''K'' of positive [[characteristic (algebra)|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]]{{Citation needed|reason=unveriviable and unsufficient citation about the source|date=May 2017}}, [[combinatorics]] and [[number theory]].
 
Within finite group theory, [[character 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 group]]s whose characterization was not amenable to purely group-theoretic methods because their [[Sylow's theorems|Sylow 2-subgroup]]s were too small in an appropriate sense. Also, a general result on embedding of elements of [[order (group theory)|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.