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