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Unformatted text preview: 4.3. 36. SYLOW THEOREMS AND APPLICATIONS 55 4.3 36. Sylow Theorems and Applications The structures of finite abelian groups are well classified. The structures of finite nonabelian groups are much more complicate (Think about S n , A n , D n , etc). Sylow theorems are very useful in studying finite nonabelian groups. Here we survey the classical results of Sylow theorems and apply them to examples. Def 4.30. Let p be a fixed prime. A group G is a p-group if every element in G has order a power of p . A subgroup of a group G is a p-subgroup of G if the subgroup is itself a p-group. Thm 4.31 (Cauchys Theorem). Let G be a finite group. Let p be a prime factor of | G | . Then G has a subgroup of order p . Cor 4.32. A finite group G is a p-group if and only if | G | is a power of p . Proof by Cauchys Theorem. If | G | is a power of p , then the order of every element g of G divides | G | . So the order of g must be a power of p . Thus G is a p-group....
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- Spring '08