course information

Mathematics 71                Fall 2005           Syllabus

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

 9-21 2.1  Definition and examples of groups p.69:  4, 5, 10, 11* and  Problems 1,2 9-23 2.2   Subgroups p. 70:  2, 3b,d,e, 7a, 11     week 1 solutions

 9-26 2.2  Cyclic subgroups and groups p.70:  10a, 12, 16(no proofs required for parts a and b.);  p. 71:  5, 6a 9-28 2.3  Isomorphisms, 1.4 permutation matrices and the symmetric group p. 71:  12                         week 2 solutions 9-30 2.4  Homomorphisms Note: 10a and 12 have been added to Monday's assignment and 11 deleted. p. 71: 14a 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*

 10-3 More 2.4 p. 71:  14b;  p. 72:  3, 6, 13*  and  Problems 3, 4 10-5 2.5  Equivalence relations p.73:  6;  p. 77:  3   and   Problems 5,6                              week 3 solutions 10-7 2.6  Cosets p. 74:  5(Hint: First show that  H intersect K is a subgroup (of H and K)), 7(Hint: Consider ker(phi)), 10   and   Problem 7

 10-10 Start 2.10  Quotient groups p. 77:  4 10-12 2.10  First isomorphism theorem p. 76:  10   and   Problems 8,9                                             week 4 solutions 10-14 2.8  Products p. 75(bottom of page):  2, 4c, 8;  p. 76:  9.8*(Just show how the version of the Chinese remainder theorem in class implies the one here.)  and   Problem 10

 10-17 Mappping property (p. 221),  5.5, 5.8  Start group actions p. 76:  11(Use the mapping property)  and   Problems 11,12 10-19 More group actions p. 194:  8.6;  p. 192: 4;  p. 193:  8(a)  and  Problem 13      week 5 solutions 10-21 More group actions, Cauchy's theorem p. 193:  4;  p. 194: 7.1(just for a tetrahedron)  and  Problems 14,15

 10-23 6.1  Class equation p. 229:  4, 6  and  Problem 16 10-25 Dihedral  groups, correspondence theorem Problems 17,18,19                        week 6 solutions 10-27 6.4 Sylow theorems p. 231:  1, 2   and   Problem 20

 10-31 More Sylow theorems take-home exam 11-2 Finite abelian groups solutions 11-4

 11-7 Uniqueness part of fundamental theorem,  start 10.1  rings week 8 solutions 11-9 10.1, 10.3 Ring homomorphims and ideals p. 379:  2(Just the anwser, no proof required);  p. 380: 13;  p. 381:  4(Also show that the ideal (2, x) is not principle.), 7  and  Problem 21 11-11 10.3 Polynomial rings p. 381:  8(b), 9, 14;   p. 382:  34

 11-14 10.4  Quotient rings p. 382:  30a,b,c    and   Problems 22,23 11-16 10.5 Adjoining elements p. 382:  3(b)(This is similar to (4.8),  p. 363), 7(a)             This week's homework  due 11-30.  week 9 solutions 11-18 More 10.5, start 10.6 p. 383:  2(Hint: Chinese remainder theorem, factor the polynomial), 6(b)(Just determine if the quotient ring is a field or an integral domain.), 8

 11-21 10.6  Intergral domains,  10.7 maximal ideals,  start factorization 11.1, 11.2 p. 384:  7.1, 7.2a,b;   p. 385: 11

 11-28 More factoring Problem 24  and  p. 384:  7.2(d)           last solutions 11-30 Euclidean domains  (p. 397) Problem 25