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Burnside's Theorem
In mathematics, Burnside's theorem in group theory states that if ''G'' is a finite group of order p^a q^b where ''p'' and ''q'' are prime numbers, and ''a'' and ''b'' are non-negative integers, then ''G'' is solvable. Hence each non-Abelian finite simple group has order divisible by at least three distinct primes. History The theorem was proved by using the representation theory of finite groups. Several special cases of the theorem had previously been proved by Burnside in 1897, Jordan in 1898, and Frobenius in 1902. John G. Thompson pointed out that a proof avoiding the use of representation theory could be extracted from his work in the 1960s and 1970s on the N-group theorem, and this was done explicitly by for groups of odd order, and by for groups of even order. simplified the proofs. Proof The following proof — using more background than Burnside's — is by contradiction. Let ''paqb'' be the smallest product of two prime powers, such that there is a non-solvable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burnside 2
Burnside may refer to: Places Antarctica * Burnside Ridges, Oates Land Australia * City of Burnside, a local government area of Adelaide, South Australia ** Burnside, South Australia, a suburb of the City of Burnside * Burnside, New South Wales, in the Oatlands suburb of Sydney * Burnside, Queensland, a suburb in the Sunshine Coast Region * Burnside, Victoria, a suburb of Melbourne * Burnside, Western Australia, in the South West region * Lake Burnside, in the Gibson Desert, Western Australia Canada * Burnside, Nova Scotia, an urban neighbourhood in Dartmouth, Nova Scotia * Burnside Drive, a road in Dartmouth, Nova Scotia * Burnside, Colchester County, an unincorporated rural community in Nova Scotia * Burnside Hall, a building on the downtown campus of McGill University, Montreal, Quebec * Burnside, Newfoundland and Labrador, a coastal community in Newfoundland * Burnside, Ontario, in the township of Severn * Burnside River, Nunavut New Zealand * Burnside, Christchur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |