Math 5310
Intr. to Modern Algebra I

• For Friday 8/27
  
  • Read Sections 1.1 and 1.2.
  • Please write me a short e-mail about yourself inclding at least the following information:
    • Your name.
    • One paragraph about your degree plans, and your previous mathematical education.
    • E-mail address for correspondence.
    • Any special situation you think I should be aware of.

• For Monday 8/30
  
  • Read Section 1.3.
  • Homework # 1 - Due on Monday 8/30
    Section 1.2, Numbers 6, 10, 13 (Problem 1.2.13 shall count as 4 problems as follows: Parts (a) through (d) shall count as one problem, and each of parts (e), (f), and (g) shall also count as one problem).

• For Wednesday 9/1
  
  • Homework # 2 - Due on Wednesday 9/1
    Section 1.3, Numbers 1, 2, 9, 10 (Problem 1.3.10 shall count as 2 problems).
    Show that, if f is a function from S to T and B is a subset of T, then the image of the inverse image of B under f is contained in B.

• For Friday 9/3
  
  • Read Section 1.4.
  • Homework # 3 - Due on Friday 9/3
    Section 1.3, Numbers 5, 7, 15, 21, 22, 27.

• For Wednesday 9/8
  
  • Read Section 1.5.
  • Homework # 4 - Due on Wednesday 9/8
    Section 1.4, Numbers 3, 5, 9, 18, 29, 34 (b)(c).

• For Friday 9/10
  
  • Homework # 5 - Due on Friday 9/10
    Section 1.5, Numbers 1, 3, 4, 5 (In number 1, all you need to do is find the greatest common divisor of the two numbers and then write it as a linear combination of them. No proof is required. For example, the book does the example of 100 and 28. All you would have to write in that case is (100, 28) = 4 and 4 = (2) * 100 + (-7) * 28).

• For Monday 9/13
  
  • Read Sections 1.6 and 1.7.
  • Homework # 6 - Due on Monday 9/13
    Section 1.5, Numbers 13, 15.
    Prove that, for every integer n, n^2-n is divisible by 2.
    Section 1.6, Numbers 1, 13.

• For Wednesday 9/15
  
  • Read Section 1.7.
  • Homework # 7 - Due on Wednesday 9/15
    Section 1.7, Numbers 2, 3, 6, 8, 10.

• For Friday 9/17
  
  • Homework # 8 - Due on Friday 9/17
    Section 1.7, Numbers 11, 16, 20, 23.

• For Monday 9/20
  
  • We are going to have our first test on Monday, September 20. You can find a list of problems to help you prepare for it in the Test Page. You can also find there a list of definitions and theorems for you to review.

• For Wednesday 9/22
  
  • Read Section 2.1.

• For Friday 9/24
  
  • Read Section 2.2.
  • Homework # 9 - Due on Friday 9/24
    Section 2.1, Numbers 1, 9, 16, 18.

• For Monday 9/27
  
  • Read Section 2.3.
  • Homework # 10 - Due on Monday 9/27
    Section 2.1, Numbers 26.
    Show that, if G is a group of order 5, then a^2 = e only if a = e.
    Section 2.2, Number 3 (Do at least the case when i=2,3).
    Section 2.3, Numbers 1, 2.

• For Wednesday 9/29
  
  • Homework # 11 - Due on Wednesday 9/29
    Section 2.3, Numbers 6, 9, 10, 11, 14.

• For Friday 10/1
  
  • Read Section 2.4.
  • Homework # 12 - Due on Friday 10/1
    Section 2.3, Numbers 16, 19, 20, 29.

• For Monday 10/4
  
  • Homework # 13 - Due on Monday 10/4
    Section 2.4, Numbers 6, 12, 16, 17.

• For Wednesday 10/6
  
  • Read Section 2.5.
  • Homework # 14 - Due on Wednesday 10/6
    Section 2.4, Numbers 9, 13, 23, 27.

• For Wednesday 10/13
  
  • Homework # 15 - Due on Wednesday 10/13
    Section 2.5, Numbers 3, 4, 15.

• For Friday 10/15
  
  • Homework # 16 - Due on Friday 10/15
    Section 2.5, Numbers 17, 18, 20, 25.

• For Monday 10/18
  
  • Homework # 17 - Due on Wednesday 10/20
    Section 2.5, Numbers 36, 41, 44.

• For Wednesday 10/20
  
  • Homework # 18 - Due on Friday 10/22
    Do 5 problems out of the the following ones:
    Section 2.3: Problems 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 26, 29.

• For Friday 10/22
  
  • Homework # 19 - Due on Monday 10/25
    Do 5 problems out of the the following ones:
    Section 2.4: Problems 6, 7, 8, 9, 12, 13, 14, 16, 17, 27, 30.

• For Monday 10/25
  
  • We are going to have our second test on Monday, October 25. You can find a list of problems to help you prepare for it in the Test Page. You can also find there a list of definitions and theorems for you to review.

• For Wednesday 10/27
  
  • Read Section 2.6.

• For Friday 10/29
  
  • Homework # 20 - Due on Monday 11/1
    Section 2.6, Numbers 1, 2, 3, 5.

• For Monday 11/1
  
  • Read Section 2.7.
  • Homework # 21 - Due on Wednesday 11/3
    Section 2.6, Numbers 7, 8, 12, 13, 17.

• For Wednesday 11/3
  
  • Homework # 22 - Due on Friday 11/5
    Section 2.7, Numbers 2, 5 (Number 5 is worth 20 points).

• For Friday 11/5
  
  • Homework # 23 - Due on Monday 11/8
    
    • Prove that Z_4 is not isomorphic to Z_2 x Z_2
    • Show that, if H and K are two finite subgroups of G such that H is contained in K, then K is the union of a finite number of cosets of H.
    • Show that, if K is a subgroup of G, then every coset Ka is the union of all cosets Kg with g an element of Ka.

• For Monday 11/8
  
  • Read Section 2.8.
  • Homework # 24 - Due on Wednesday 11/10
    
    • Section 2.7, Number 6.
    • Section 2.8, Number 7.
    • Show that Z_6 is isomorphic to Z_2 x Z_3
    • Prove that every group of order 99 which contains an abelian normal subgroup of order 9 is abelian.

• For Wednesday 11/10
  
  • Homework # 25 - Due on Friday 11/12
    Section 2.8, Numbers 2, 10, 11.

• For Friday 11/12
  
  • Homework # 26 - Due on Monday 11/15
    Section 2.9, Numbers 5, 6 (Do both of them just for k=3).

• For Monday 11/15
  
  • Homework # 27 - Due on Wednesday 11/17
    Section 2.10, Numbers 1, 2, 3.

• For Wednesday 11/17
  
  • Homework # 28 - Due on Friday 11/19
    
    • Section 2.9, Number 2.
    • Classify all abelian groups of order 12 up to isomorphisms.
    • Classify all abelian groups of order 16 up to isomorphisms.
    • Classify all abelian groups of order 42 up to isomorphisms.

• For Friday 12/19
  
  • Homework # 29 - Due on Monday 11/22
    Do 5 problems out of the the following ones:
    Section 2.6: Problems 1, 2, 3, 4, 5, 7, 8, 11, 12, 13, 15, 17.

• For Monday 11/22
  
  • We are going to have our third test on Monday, November 22. You can find a list of problems to help you prepare for it in the Test Page. You can also find there a list of definitions and theorems for you to review.

• For Wednesday 12/1
  
  • Homework # 30 - Due on Friday 12/3
    Section 3.2, Numbers 2, 3, 10, 11, 15.

• For Friday 12/3
  
  • Homework # 31 - Due on Monday 12/6
    Section 3.2, Number 20.
    Section 3.3, Numbers 1, 3, 5.

• For Monday 12/6
  
  • Homework # 32 - Due on Monday 12/6
    Write up the 13 definitions you need to know to prepare for the Final, i.e., the 13 definitions from the preparation for Tests #2 and #3.

• For Wednesday 12/8
  
  • Homework # 30 - Due on Wednesday 12/8
    Do any 5 problems from the preparation for the Final.

Last updated on December 15, 2004.