Professor: Jeff Hoffstein
Office: 104 Kassar-Gould House
Email: jhoff@math.brown.edu
Office Hours: Wednesday 1-4
| Calendar | Material Covered | Homework Assignments |
| Jan 25
- Jan 27 | Chapter 1 (Introduction to groups) | See below for HW #1 |
| Jan 30 - Feb 3 | Chapter 1 continued | See below for HW #2 |
| Feb 6-8
Feb 10 | Chapter 1 continued Chapter 2 (Subgroups) | See below for HW #3 |
Feb 13- Feb 17 | Chapter 2 continued | See below for HW #4 |
| Feb 20- 22 Feb 24 | Chapter 3 (Quotient groups and homomorphisms) | See below for HW #5 |
Feb 27 - March 2 | Chapter 3 continued | See below for HW #6 |
March 5 | Chapter 3 continued (Through Section 3.3) | No HW |
| March 12- March 16 | Chapter 4 continued | See below for HW #7 |
| March 19 March 21-23 | Chapter 4 continued | See below for HW #8 |
| Spring break | ||
| April 2-4 April 6 | Chapter 4 continued, Chapter 7 (Introduction to rings) | No HW |
| April 9-13 | Chapter 7 continued | See below for HW #9 |
| April 16-20 | Chapter 7 continued
| See below for HW #10 |
| April 23-reading period
| Selections from Chapters 8,9,13
| See below for HW #11 |
Challenge problems don't need to be done or handed in. They are there for those interested in doing them. And come tell me about the solutions.
HW#1: Section 1.1: 1a,b ; 2 a,b; 5, 6, 7, 8, 14, 22, 25; Section 1.2: 2, 3, 10
There's a lot here but it's really important to drill this stuff into your brain, both by just plain mechanical exercises, and by thoughtful excercises. It's a good idea to do as many other excercises as you can stand to do also.
Challenge: If G is a group in which (a b)^i = a^i b^i for three consecutive integers i, for all a ,b in G show that G must be abelian (commutative) (First check that if it's true for i = 2 it's easy.)
Challenge: Suppose G is a non-empty set, closed under an asociative product operation. Suppose also that
a) There exists e in G such that a e = a for all a in G
b) Given any a in G there exists y(a) in G such that a y(a) = e.
Show that G must be a group. Show by example that if you replace b) by y(a) a = e then G might not be a group.
HW#2: Section 1.3: Do 2,3 just for sigma tau and tau sigma; 10; Section 1.4: 3,4,11(a,b,c); Section 1.6: 2,3,14,17,26
Problem #26 (courtesy of Daniel Parker)
HW#3: Section 1.7: 2,4,16, 17, 18,19; Section 2.1: 3,5,8, 11 (see excercise 28 in section 1.1 for definition of A x)
HW#4: Section 2.2: 2 ,4, 6, 11; Section 2.3: 1, 3, 12, 24, 26; Section 2.4: 7,13,14,15;
HW#5: Section 3.1: 1, (2), 3,4, (5), 9, 10, (12), 18, 22, 24, (27,31,34), 36, (41)
HW#6: Section 3.2: 4, 7, 8, 12, 18, 19, 22; Section 3.3: 3, (4), 6, (7)
Challenge: Suppose that G is a finite abelian group, in which the number of solutions in G of x^n = e is at most n for every positive integer n. Prove G must be a cyclic group.
Challenge: Suppose G is an infinite group and H, K are two subgroups of G that are of finite index in G. Prove that the intersection of H and K is also of finite index. Can you find an upper bound for it, in terms of [G:H] and [G:K]?
HW#7: Section 4.1: 1; Section 4.2: 2,4, 7,8,10; Section 4.3: 2a,b, 5,6
HW#8: Section 4.4: 6, 13; Section 4.5: 12,16,17
(and possibly a few more from 4.4 and 4.5)
HW#9 Section 7.1: 3,11,24, Section 7.2: 10,b,c, Section 7.3: 2,3,8,17,22,26,28,34