course information

Mathematics 71                Fall 2004           Syllabus

Date             Topics                                                                                 Homework (Do not hand in the starred problems.)

 9-22 2.1  Definition and examples of groups p.69:  4, 5, 10, 11* and  Problems 1,2 9-24 2.2   Subgroups p.70:  2, 3c,d,e, 7a, 11      week 1 solutions

 9-27 2.2  Cyclic subgroups and groups p.70:  10(a), 12, 16 (parts (b) and (c) are optional.) 9-29 2.3  Isomorphisms p.71:  5, 6*, 12a, 14ab, 16*                      week 2 solutions 10-1 2.4  Homomorphisms, 1.4 permutation matrices and the symmetric group p. 35:  1(In part (b), just write p as a product of transpositions.), 2(Also prove that every permutaion is a product of transpositions), 4;  p. 72: 2*

 10-4 More 2.4,  Start: 2.5  Equivalence relations p.72:  3, 6, 7*, 10* and  Problems 3,4 10-6 More 2.5,  2.6 cosets p.77:  3   and   Problems 5,6,7 10-8 More 2.6,  2.10 quotient groups p.77:  4;  p.74:  7, 10, 12*             week 3 solutions

 10-11 2.10 First isomorphism theorem,  Start 2.8 products p. 76:  10.5, 10.10*  and  Problems 8,9 10-13 2.8 Products p.75:  2, 3*, (4ac)*, 8;   p.76:  9.8*  and  Problem 10 10-15 Mapping properties p. 75:  11(a)  and  Problems 11,12             week 4 solutions

 10-18 5.5, 5.8  Start group actions p.194: 8.6;  p.192:  4  and  Problem 13 10-20 5.6, 2.7  More group actions p. 193:  5.8*, 6.1*, 6.4;  p. 194: 7.1(just for a tetrahedron)  and  Problems 14,15 10-22 Cauchy's Theorem,  Start 6.1 Class equatation p. 194:  8.4*;   p. 229:  4*, 6, 10(e)   and   Problem 16      week 5 solutions

 10-25 Dihedral groups, start correspondence theorem Problems 17,18,19 10-27 Start Sylow theorems p.231:  1, 2   and   Problem 20 10-29 6.4  Sylow theorems Problem 21                   week 6 solutions

 11-1 Start semidirect products take-home exam 11-3 solutions 11-5 6.5  Groups of order 12

 11-8 Start  10.1 Rings p. 379: 1b,c;   p. 380: 12*, 13, 14* 11-10 10.3 Homomorphisms and ideals p. 381: 4(Also show that the ideal (2, x) is not principle.), 7, 8(b)(What is a generator for the kernel?)  and  Problem 22             week 8 solutions 11-12 10.3 Polynomial rings p, 381: 9, 14;   p. 382:  34

 11-15 10.4 Quotient rings p. 382:  3(b)(This is similar to (4.8), p. 363), 7(a)  and  Problem 23 11-17 10.5 Adjoining elements p. 383:  6(b), 8, 9                 week 9 solutions 11-19 10.6 Integral domains  10.7 Maximal ideals p. 383: 2(Hint: Chinese remainder theorem);  p. 384: 7.2(a)   and  Problem 24

 11-22 11.1, 11.2  Start factorization p. 384: 7.1, 7.2bd;   p. 385: 11;  p.442: 9(a)*  and  Problem 25

 11-29 More Factoring Problem 26                 last solutions 11-31 Euclidean domains, Gauss lemma Homework assigned 11-22 and after is optional.