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Burnside's Theorem

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https://ir.library.oregonstate.edu/concern/graduate_projects/5x21tg13g

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  • Just as prime numbers can be thought of as the building blocks of the natural numbers, in a similar fashion, simple groups may be considered the building blocks of finite groups. Burnside considered the following questions: 1. Do there exist non-abelian simple groups of odd order? 2. Do there exist non-abelian simple groups whose orders are divisible by fewer than three distinct primes? In 1904, Burnside answered question 2 when he used representation theory to prove that groups whose orders have exactly two prime divisors are solvable. His proof is a clever application of representation theory, and while purely group-theoretic proofs do exist, they are longer and more difficult than Burnside's original proof. This paper presents a representation theoretic proof of Burnside's Theorem, providing sufficient background information in group theory and the representation theory of finite groups, and then gives a brief outline of a group theoretic proof.
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