Fraleigh - Incomplete Notes on Fraleigh’s Abstract...

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Unformatted text preview: Incomplete Notes on Fraleigh’s Abstract Algebra (4th ed.) Afra Zomorodian December 12, 2001 Contents 1 A Few Preliminaries 2 1.1 Mathematics and Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Sets and Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Complex and Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Introduction to Groups 4 2.1 Binary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Groups of Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Orbits, Cycles, and the Alternating Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.6 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.7 Cosets and the Theorem of Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.8 Direct Products and Finitely Generated Abelian Groups . . . . . . . . . . . . . . . . . . . . . 9 3 Homomorphisms and Factor Groups 11 3.1 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Isomorphism and Cayley’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Factor Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Factor-Group Computations and Simple Groups . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.5 Series of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.6 Groups in Geometry, Analysis, and Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Advanced Group Theory 16 4.1 Isomorphism Theorems: Proof of the Jordan-H¨ older Theorem . . . . . . . . . . . . . . . . . . 16 4.2 Group Action on a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3 Application of G-Sets to Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.4 Sylow Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.5 Applications of the Sylow Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.6 Free Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.7 Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.8 Group Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1 5 Introduction to Rings and Fields...
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