**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.

Lectures |

Problem sets |

Exams |

Vocabulary list |

- 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

- 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; M
_{2}denotes 2x2 matrices under addition, GL_{2}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*to_{m}*S*such that some odd permutations of_{n}*S*are mapped to even permutations of_{m}*S*._{n}

(b) For any positive*m*, show that*A*is generated by cycles of length 3 (i.e., every element may be written as a product of cycles of length 3)._{m}

(c) Fix*m< n*arbitrary positive integers. Show that for any homomorphism*S*to_{m}*S*, every even permutation of_{m}*S*is mapped to an even permutation of_{m}*S*._{m} - Problem set 4, due 3/3: Section 7.1, exercise 1 (note that
(
**Z**/32**Z**)^{×}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

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.

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.

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 |