Mathematics 5331  Abstract Algebra I Schedule/Homework

Spring 2009

 Date Section/Topic Homework Tuesday 1/20 §1.1 Semigroups, monoids and groups Problem 4, p.29. Give the Cayley table for the group of symmetries of the square.  Problem 5, p.29. Prove the order of Sn is n!  Prove Sn is non-abelian for n greater than 2.  Problems 13, 14 pp. 30 Thursday 1/22 § 1.1 Properties of Groups Prove Proposition 1.3 on page 25. Problem 6 on page 29. Prove Proposition 1.4 on page 25. Problem 15 on page 30. Tuesday 1/27 UTA CLOSED Thursday 1/29 § 1.2 Homomorphisms and subgroups Pages 33-34 Problems # 1, 2, 6, 7, 9, 10, 11, 13 HW # 1: Turn-in Problems # 6 and # 11  on Tuesday February 10th Tuesday 2/3 § 1.3 Cyclic groups Pages 36-37 Problems # 1, 3, 4, 6, 8,  9 HW # 2: Turn-in Problem # 3 on Thursday February 12. Thursday 2/5 NO CLASS Tuesday 2/10 § 1.4 Cosets and counting Page 40 # 2, 3, 6 Thursday 2/12 § 1.5 Normality Page 45 # 1, 2, 5, 6, 7, 9a HW # 3: Turn-in Problems: # 6 page 40; # 1, 7 page 45  on Thursday 2/19. Tuesday 2/17 § 1.5 Quotient groups Problems in group theory (Handout I ) Thursday 2/19 Test # 1 Tuesday 2/24 § 1.5 Isomorphism Theorems Pages 45-46 #  11, 16 HW # 4: Turn-in Problems: # 11, 16 pages 45-46  on Thursday 3/5. Thursday 2/26 § 1.6 Symmetric group Study the proof of Corollary 6.4. Page 51 # 2, 3, 4 Tuesday 3/3 § 1.6 Alternating group. Cayley's Theorem Page 51 # 5, 8 HW # 5: Turn-in Problems pp. 51-52 # 3, 5 on Thursday 3/12 Thursday 3/5 § 1.8 Direct products. § 2.2 Finitely generated abelian groups Handout (Chapter 5) p. 156 # 1; p.165 # 1 a, b; 2 a, b, c Handout problems on group Theory # 3-7 Tuesday 3/10 § 2.2 Invariant factors. Elementary divisors Handout (Chapter 5)  p.165 # 3 a, b, c Handout Problems on Group Theory # 21-25, 29 HW # 6: Turn-in Problem # 12 p.82  on Tuesday 3/24 Thursday 3/12 § 2.4 Group actions March 16-20 Spring break Tuesday 3/24 § 2.4 Group actions Page 92 # 3, 9, 14          Handout Problems in Group Theory II-All Thursday 3/26 Test # 2 Tuesday 3/31 § 2.5 Sylow's Theorems Problems # 1, 10, 11, 13 on page 96 Thursday 4/2 § 2.5 Sylow's Theorems 1. Show that a group of order 36 is not simple.2. Handout - Problems on Group Theory-Problems # 1- 6 3. Handout-Problems on Sylow Theorems-# 1-10  HW # 7. Turn in Problems # 1, 10, 11, 13 on page 96 on Thursday 4/9 Tuesday 4/7 § 2.6 Finite groups Study the proof of Proposition 6.3 and Proposition 6.4See link below for a listing of groups of small order. http://www.math.usf.edu/~eclark/algctlg/small_groups.html Pages 99-100  # 3, 4, 10  HW # 8. Turn in Problems # 1,3, 5 from Handout Problems in Group Theory and problems # 3, 4 from page 99 on Thursday 4/16 Thursday 4/9 § 2.7 Solvable groups. Pages 106-107 # 2, 10, 14 Tuesday 4/14 § 2.7 Nilpotent groups Thursday 4/16 § 3.1 Rings and homomorphisms HW # 9. Due Thursday 4/23       Page 120 # 3, 6,  7, 15 Tuesday 4/21 Review Test # 2 Thursday 4/23 Test # 2 Tuesday 4/28 § 3.2 Ideals. The Isomorphism Theorems Page 133 # 3, 4 Thursday 4/30 § 3.2 Quotient rings. The Isomorphism Theorems HW # 10. Due Thursday 5/7      Page 133 # 3, 4, 7a, 10 Tuesday 5/5 § 3.2 Prime and maximal ideals Page 134 # 20 Thursday 5/7 §3.3 Factorization in commutative rings Page 140 # 1 Tuesday 5/12 Final Examination 11:00 a.m.-1:30 p.m.