Group Actions, p-Groups, and the Sylow Theorems
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چکیده
In this note we introduce the notion of a group action on a set and use it to prove a number of theorems about p-groups and the Sylow theorems. For undefined terms see any decent book on group theory. The theory of p-groups and the Sylow theorems have a number of applications in Galoistheory. The nice structure of p-groups will translate via the fundamental theorem of Galois theory to nice structure of Galois extensions of degree a power of a prime. An application of the Sylow theorems to Galois theory is to give a mostly algebraic proof that C is algebraically closed.
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Course 311: Michaelmas Term 2005 Part II: Topics in Group Theory
2 Topics in Group Theory 2 2.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Examples of Groups . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Elementary Properties of Groups . . . . . . . . . . . . . . . . 4 2.4 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 Co...
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