Homework


Homework 1 - Due Wed. Jan. 28

Todo:
  • Read the syllabus.
  • Read the following advice for students taking Modern Algebra by Joe Gallian.
  • Read the following advice on proof writing (also by Joe Gallian).
  • Some of you will also find the following interesting: reasons why Modern Algebra is valuable to math education (and math) majors. (also by Joe Gallian).

    • To turn in:
  • Complete the intro sheet.

    • Homework 2 ---- Part 1 Due Mon. Feb. 2 ---- Part 2 Due Wed. Feb 4

    Read Chapters 1 and 2.

    Part 1: Read Chapters 1 and 2, then complete Chapter 1: 18,19,21,22,24,25,28,29.
  • Group 1 will present: 18.
  • Group 2 will present: 21,25.
  • Group 3 will present: 24a,24b.
  • Group 4 will present: 24c,24d.
  • Group 5 will present: 29.

    • Part 2: Chapter 2: 4,5,16,18(use 16),22,27

    You will always turn in all of Part 2 and any problems from Part 1 not presented. For this week, Chapter 1: 19,22,28 and Part 2 above..

    Extra Fun: Chapter 2: 8,21

    Homework 3 ---- Part 1 Due Mon. Feb. 9 ---- Part 2 Due Wed. Feb 11

    Read Chapter 3.

    Part 1: Read Chapter 3, then complete Chapter 3: 1,2,4,6,7,10,14,26.
  • Group 1 will present: 4.
  • Group 2 will present: 7 (make sure you show the steps for associativity).
  • Group 3 will present: 2.
  • Group 4 will present: 14.
  • Group 5 will present: 10.

    • Part 2: Chapter 3: 27(use induction),30,37,38(only prove this is a subgroup),39,40

    Extra Fun: Chapter 3: 31,33,38(prove the rest),47,48,54,55

    Exam 1 - Wed. Feb. 18 - Covers Homeworks 2-3 (work out the extra problems)

    Homework 4 ---- Part 1 Due Mon. Feb. 23 ---- Part 2 Due Wed. Feb 25

    Read Chapter 4.

    Part 1: Read Chapter 4, then complete Chapter 4: 1(a-d),2,3(a,b,d,h,j,k),4(a-d),5,8,9,10,11,12.
  • Group 1 will present: 4(a-d).
  • Group 2 will present: 2,5.
  • Group 3 will present: 3(a,b,d,h,j,k).
  • Group 4 will present: 8.
  • Group 5 will present: 1.

    • Part 2: Chapter 4: 20,23,24,25,26,31,37,39,42

    Extra Fun: Chapter 4: 1e,17,18,19,21,27

    Homework 5 ---- Part 1 Due Mon. Mar. 2 ---- Part 2 Due Wed. Mar. 4

    Read Chapter 5.

    Part 1: Read Chapter 5, then complete Chapter 5: 1,2(e-o),3,4,5(just parts a-c and the question after),7,8,10(only up to n=6), Start Part 2!.
  • Group 1 will present: 3,4.
  • Group 2 will present: 7,8.
  • Group 3 will present: 1,2(e-o).
  • Group 4 will present: 5(just parts a-c and the question after).
  • Group 5 will present: 10(only up to n=6).

    • Part 2: Chapter 5: 11,14,17,18,24,27,31,34,35

    Extra Fun: Chapter 5: 10(for remaining n=7,8,9,10),36

    Homework 6 ---- Part 1 Due Mon. Mar. 9 ---- Part 2 Due Wed. Mar. 11

    Read Chapter 6.

    Part 1: Read Chapter 6, then complete Chapter 6: 5 (Hint: Use the properties of cosets to cut the work down, for example property 4 should help), Prove the following properties from the Properties of Cosets in the lecture notes: Property 4, Property 5, Property 6.
  • Group 1 will present: 5c (but use \(4\mathbb{Z}\) instead of \(3\mathbb{Z}\), also try \(n\mathbb{Z}\) in general), Property 4
  • Group 2 will present: 5g, Property 5
  • Group 3 will present: 5d, Property 6
  • Group 4 will present: 5a, 5b, Consider the group \(G=(\mathbb{R}^{*},\cdot)\), subgroup \(H=\langle -1\rangle\), and let \(x\in G\); what is the left coset \(xH\)?
  • Group 5 will present: 5f, 6 (ignore the index part of the question for now)

    • Part 2: Chapter 6: 1,2,12, Complete these problems

    Extra Fun: Chapter 6: 5e,5h,16,17,18,19

    Homework 7 ---- Part 1 Due Mon. Mar. 30

    Read 9.1.

    Part 1: Read Chapter 9.1, then complete these problems. Work on these with your group and we can share solutions on Monday.
    Part 2: This will be given after the Exam.

    Exam 2 - Wed. Apr. 1 - Covers Homeworks 4-7

    Homework 7 ---- Part 1 Due Mon. Apr. 6 --- Part 2 Due Wed. Apr. 8

    Read 9.1.

    Part 1: Read Chapter 9.1, then complete these problems. Work on these with your group and we can share solutions on Monday (for real this time!).

    Remember: It is usually easier to show two groups are NOT isomorphic by finding some essential property the groups fail to share. e.g., only one is cyclic, abelian, has element of order 3, etc.
    Part 2: Chapter 9: 2,3,4,5,6,8 (Recall Ch. 4#1c - \(\mathbb{Q}\) is not cyclic ),34,35,36

    Extra Fun: Chapter 9: 41 Challenge: 11,12,14

    Homework 8 ---- Part 1 Due Mon. Apr. 13 --- Part 2 Due Wed. Apr. 15

    Read 9.2.

    Part 1: Read 9.2, then complete Chapter 9:.
  • Group 1 will present: 52
  • Group 2 will present: 15,16
  • Group 3 will present: 48
  • Group 4 will present: 47
  • Group 5 will present: 50

    • Part 2: Complete these problems.

    Extra Fun: Chapter 9: 31 Challenge: 37-42

    Homework 9 - Due Wed. Apr. 22

    Read 10.1.

    Complete These problems, and then:
    Complete Chapter 10: 5,6,8.

    Challenge Problems and Extra Fun: Chapter 10: 7,9-14

    Exam 3 - Wed. Apr. 29 - Covers Homeworks 7-9

    Homework 10 - This will not be graded, but you should do these!

    Read 11.1,11.2,Theorem 13.3,16.

    Chapter 11: 1,2,3,4,8,9,19
    Chapter 13: 1,2,3
    Chapter 16: 1,2,3

    Challenge Problems Ch11: 5-7,10,11,15-17
    Challenge Problems Ch16: 10,17,18,32,34,38