TAILIEUCHUNG - GROUP THEORY

Abstract. Thes are the notes for the first part ofMath 594, University ofMichigan,Winter 1994, exactly as they were handed out during the course except for some minor corrections. Please send comments and corrections to me at jmilne@ using “Math594” as the subject. | GROUP THEORY . MILNE August 21 1996 Abstract. Thes are the notes for the first part of Math 594 University of Michigan Winter 1994 exactly as they were handed out during the course except for some minor corrections. Please send comments and corrections to me at jmilne@ using Math594 as the subject. Contents 1. Basic Definitions 1 . Definitions 1 . Subgroups 3 . Groups of order 16 4 . Multiplication tables 5 . Homomorphisms 5 . Cosets 6 . Normal subgroups 7 . Quotients 8 2. Free Groups and Presentations 10 . Free semigroups 10 . Free groups 10 . Generators and relations 13 . Finitely presented groups 14 The word problem The Burnside problem Todd-Coxeter algorithm Maple 3. Isomorphism Theorems Extensions. 16 . Theorems concerning homomorphisms 16 Factorization of homomorphisms The isomorphism theorem The correspondence theorem Copyright 1996 . Milne. You may make one copy of these notes for your own personal use. i ii . MILNE . Products 17 . Automorphisms of groups 18 . Semidirect products 21 . Extensions of groups 23 . The Holder program. 24 4. Groups Acting on Sets 25 . General definitions and results Orbits Stabilizers Transitive actions The class equation p-groups Action on the left cosets 25 . Permutation groups 31 . The Todd-Coxeter algorithm. 35 . Primitive actions. 37 5. The Sylow Theorems Applications 39 . The Sylow theorems 39 . Classification 42 6. Normal Series Solvable and Nilpotent Groups 46 . Normal Series. 46 . Solvable groups 48 . Nilpotent groups 51 . Groups with operators 53 . Krull-Schmidt theorem 55 References Dummit and Foote Abstract Algebra. Rotman An Introduction to the Theory of Groups GROUP THEORY 1 1. Basic Definitions . Definitions. Definition . A is a nonempty set G together with a law of composition a b a b G X G G satisfying the following axioms a associative law for all a b c E G a b c a b c b existence of an identity .

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• Subgroups
• Groups of order
• Multiplication tables
• Homomorphisms
• Cosets
• Normal subgroups
• Quotien
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