Math H113 - Honors Introduction to Abstract Algebra

Instructor: Brian Osserman

Lectures: MWF 2:00-3:00pm, Room 85 Evans

Course Control Number: 54927

Office: 767 Evans, e-mail:

Office Hours: Tu 2:00-3:00pm, W 1:00-2:00pm, F 3:00-4:00pm (starting 1/24)

GSI: Sami Assaf

GSI Office Hours: WTh 9:30am-12:00pm, 1:00pm-3:30pm, in 891 Evans

Prerequisites:

Required Text: Beachy/Blair, Abstract Algebra, 3rd edition

Recommended Reading:

Syllabus: We will hopefully cover groups, rings, and fields, with some exploration of Galois theory at the end to tie concepts together.

Grading: 45% homework, 15% x 2 exams, 25% final.

Homework: Homework will be assigned weekly, due on Fridays.

Quick links
Lectures
Problem sets
Exams
Vocabulary list

Lectures

The parts of the textbook discussed in each lecture will be posted here.

Lecture 1, 1/18: Introduction and overview of the course
Lecture 2, 1/20: Rapid overview of chapters 1 and 2 of the book, except for sections 1.2 and 2.3.
Lecture 3, 1/23: First portion of section 2.3
Lecture 4, 1/25: Rest of section 2.3, section 3.1
Lecture 5, 1/27: First portion of section 3.2
Lecture 6, 1/30: Rest of section 3.2
Lecture 7, 2/1: Section 3.8
Lecture 8, 2/3: Part of section 3.7, section 3.4
Lecture 9, 2/6: More of section 3.7, application to cyclic and permutation groups
Lecture 10, 2/8: Rest of section 3.7
Lecture 11, 2/10: Cyclic groups, dihedral groups, alternating groups (see sections 3.5, 3.6)
Lecture 12, 2/13: Product groups (see section 3.3), review for exam
Lecture 13, 2/17: First portion of section 7.1
Lecture 14, 2/22: Rest of section 7.1
Lecture 15, 2/24: Most of section 7.3
Lecture 16, 2/27: Rest of section 7.3, section 7.2
Lecture 17, 3/1: Most of section 7.4
Lecture 18, 3/3: Rest of section 7.4, beginning of section 7.5
Lecture 19, 3/6: Rest of section 7.5, comments on classification of finite simple groups
Lecture 20, 3/8: Section 5.1
Lecture 21, 3/10: Most of sections 5.2 and 5.3
Lecture 22, 3/13: Rest of sections 5.2 and 5.3
Lecture 23, 3/15: Section 5.4
Lecture 24, 3/17: Roots of polynomials over fields, squares modulo p
Lecture 25, 3/20: Every prime congruent to 1 mod 4 is a sum of squares
Lecture 26, 3/24: Part of section 6.1
Lecture 27, 4/4: Rest of section 6.1, part of 6.2
Lecture 28, 4/5: Rest of section 6.2
Lecture 29, 4/7: Section 6.4
Lecture 30, 4/10: First half of Section 6.3
Lecture 31, 4/12: Second half of Section 6.3
Lecture 32, 4/14: First portion of Section 8.1
Lecture 33, 4/17: Rest of Section 8.1, first part of Section 8.2
Lecture 34, 4/19: Rest of Section 8.2
Lecture 35, 4/21: First portion of Section 8.3

Upcoming lectures: 8.3, 7.6, 8.4

Problem sets

Problem sets will be posted here over the course of the semester. Problems sets will be due by 4:00 pm on Fridays unless marked otherwise. Routine late problem sets will be accepted only at the discretion of the grader, but in exceptional circumstances I will grant extensions upon request.

No homework due 1/27, but make sure you are comfortable with the material discussed in lecture on 1/20.

Problem set 1, due 2/3: Section 2.3, exercises 3 (omit associated diagram), 13; 3.1, ex. 16, 20, 21; 3.2, ex. 9, 18

Problem set 2, due 2/10: Section 3.2, exercises 15, 25; 3.8, ex. 5, 24; 3.7, ex. 7 (justify your answers; M₂ denotes 2x2 matrices under addition, GL₂ denotes invertible 2x2 matrices under multiplication), 9

No homework due 2/17, but look at the practice exam below.

Problem set 3, due 2/24: Section 3.6, exercises 19, 21; 3.8, ex. 25. In addition, go through and work out the entire exam 1 (you do not need to turn this in), and do the following problem:
(a) Give an example of some m< n and an injective map from S_m to S_n such that some odd permutations of S_m are mapped to even permutations of S_n.
(b) For any positive m, show that A_m is generated by cycles of length 3 (i.e., every element may be written as a product of cycles of length 3).
(c) Fix m< n arbitrary positive integers. Show that for any homomorphism S_m to S_m, every even permutation of S_m is mapped to an even permutation of S_m.

Problem set 4, due 3/3: Section 7.1, exercise 1 (note that (Z/32Z)^× is the group of equivalence classes mod 32 of odd numbers, under multiplication), 9 and find an example of a group G such that G/Z(G) is a non-trivial abelian group; 7.3, ex. 1, 2, 4

Problem set 5, due 3/10: Section 7.2, exercises 4,9; 7.3, ex. 6,12; 7.5, ex. 11, 12

Problem set 6, due 3/17: PDF here.
No homework due 3/24, but look at the practice exam below.
Problem set 7, due 4/7: carefully write up solutions to both Exam #1 and Exam #2.
Problem set 8, due 4/14: Section 6.1, exercises 7,9; 6.2, ex. 4,7,11; 6.4, ex. 4
Problem set 9, due 4/21: Section 6.4, exercises 7,10; 8.1, ex. 7; 8.2, ex. 4
Problem set 10, due 4/28: Section 8.1, exercises 5,6; 8.2, ex. 5,8 (note that GF(p)=Z/pZ); 8.3, ex. 3,7
Problem set 11, due 5/5: Section 8.4, exercises 6,7,8; 7.6, ex. 7,9

Exams

The first exam will be on 2/15, so that grades can be returned before the drop deadline, which is 2/17. It will be a one-hour, in-class, closed-book exam. Bring a blue book.

Practice exam #1 Note: (1) is longer than you should expect on the exam, and (3) is substantially longer and more difficult. If you replace $S_4$ by $S_3$ in (3), the problem would be more in line with what to expect on the exam.

Solutions to practice exam #1.

The second exam will be on 3/22, with grades returned by 3/24. As before, one-hour, in-class, closed-book, bring your own blue book.

Practice exam #2

Solutions to practice exam #2.

The final exam will be from 5-8 PM, Saturday 5/13, in 85 Evans. The format will be the same as the previous exams, with 8-10 problems instead of 4. The exam will be comprehensive, but weighted toward material covered since spring break (i.e., fields and Galois theory). Don't forget blue books.

Vocabulary list

A list of terminology which, for one reason or another, will be different in class from in the book.

In lecture	In the book
Injective [function]	One-to-one [function]
Surjective [function]	Onto [function]
Bijective function	One-to-one correspondence
Quotient [set, group,ring]	Factor [set, group,ring]
Field of fractions	Quotient field