We give five proofs of Sylow's Theorem. The proofs given will be Sylow's original proof, the Cauchy-Frobenius proof using double cosets, Frobenius' proof (two versions) using the class equation, and Wielandt's proof using subsets. We give a number of different examples to illustrate the various proofs. We put Sylow's work in historical context. We give an application of Sylow's Theorem in the proof of the Fundamental Theorem of Algebra. We give a determination of conjugacy classes in the Dihedral group. We conclude with a sketch of Sylow.
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