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.