## Math 71: Abstract Algebra

### Fall 2015

General Info
Diary

Policy: homework should be submitted no later than 5pm on the due date. Let your instructor know in advance if you feel that you need extra time.
In order to be well-prepared for the examinations, it is strongly suggested that you treat all the problems listed here.

 wk date reading topic homework / comments 1 09/16 §0.1 - §0.3 Equivalence relations, integers modulo n §0.1: 1, 4, 7§0.2: 1, 2, 3, 7, 10, 11§0.3: 4, 6, 10, 11, 12, 13, 14 09/17 (x) §1.1 Groups: definitions and first examples §1.1: 1, 2, 5, 6, 8, 9, 11, 13, 14, 17, 20, 22, 23, 24, 25, 29, 30, 34 09/18 §1.4, §1.5 Examples of groups §1.4: 1, 2, 4, 5, 8, 10, 11§1.5: 2 2 09/21 § 1.2 Dihedral groups §1.2: 1(a,b), 2, 3, 7, 9, 10, 14, 15, 17, 18 09/23 § 1.3, §1.6 Symmetric groups, group homomorphisms §1.3: 1, 4(a), 5, 6, 9(a), 11, 13§1.6: 1 - 4, 8, 9, 13, 16, 17, 19, 20, 25 09/24 (x) Problem session Homework #1, due Tuesday 09/29 and solution. 09/25 §2.1 Subgroups §2.1: 1 to 17 3 09/28 §2.3 Cyclic groups §2.3: 1 - 3, 9, 12, 13, 16, 18, 20, 21, 26 09/30 §2.4 Subgroups generated by subsets §2.4: 1 - 6, 13, 14, 18, 20, 21, 26 10/01 (x) Problem session Homework #2, due Tuesday 10/06 and solution. 10/02 §3.1 Quotients, First Isomorphism Theorem §3.1: 1 - 3, 5, 7, 9, 10, 11, 12, 13, 14, 16, 22, 28, 29, 35, 36, 37, 38, 40, 41 §3.2: 1, 4, 5, 6, 7, 8, 9, 11, 12, 15, 16, 18, 19, 22§3.3: 1, 3, 4, 8, 9 4 10/05 Problem session 10/07 §3.2 Lagrange's Theorem 10/08 Midterm 1 and solution Homework #3, due 10/16 and solution. 10/09 §3.5 Transpositions and the Alternating Group §3.5: 1 - 4, 6, 9, 14 5 10/12 §1.7, §4.1 Group actions §1.7: 1 - 4, 6, 8, 13, 16, 17, 19, 21 §4.1: 1 - 3, 6, 7, 8, 10 10/14 §4.2 Group actions §4.2: 8, 10, 14 10/16 §4.3 Group acting by conjugation §4.3: 4, 5, 8, 23, 24, 25 6 10/19 §3.4, §4.5 Composition series and the Hölder Program, Sylow's Theorems §3.4: 1, 4, 5, 7§4.5: 2, 3, 10, 17, 27, 32 10/21 §5.2, § 5.4 The Fundamental Theorem of Finitely Generated Abelian groups, Direct Products §5.2: 1, 5, 6, 7, 11, 14§5.4: 1 - 5, 15, 16, 20 10/22 (x) §5.5 Semi-direct products §5.5: 1, 2, 3, 18, 23 10/23 §7.1, § 7.2 Rings: polynomial, matrix and group rings §7.1: 1 - 7, 11, 12, 13, 23, 25, 26, 27§7.2: 1, 3, 5, 6, 7, 8, 9, 12 7 10/26 §7.3 Ring homomorphisms §7.3: 1 - 4, 6, 7, 8, 10, 11, 12, 13, 16, 17, 23, 24, 25, 26, 28, 29, 34 10/28 §7.4 Ideals §7.4: 1 - 6, 10, 11, 14, 18, 19, 29, 30, 31, 33, 38 10/29 Midterm 2 and solution Take-home and solution Homework #4, due Friday 11/06 and solution. 10/30 §8.1, §9.1 Euclidean domains, polynomial rings §8.1: 1 - 5, 7, 8, 10§9.1: 1 - 9, 13, 15, 16, 17, 18 8 11/02 §8.2, §9.2 Principal ideal domains §8.2: 1 - 8§9.2: 1 - 12 11/04 §8.3 Unique factorization domains §8.3: 1 - 9, 11 11/05 (x) Problem session Homework #5, due Monday 11/16 and solution. 11/06 §8.3 Unique factorization domains 9 11/09 §9.3 Gauss Lemma and consequences §9.3: 1 - 5 11/11 §9.3 Transfer theorem for UFDs 11/12 (x) Problem session 11/13 §13.1 Extension fields 10 11/16 Further topics Introduction to representation theory Final examination, due 11/24 at 8am and solution.