Math 31, Winter 2005

Topics in Algebra

Homework assignments

Homework assignments are to be handed in weekly. In general, assignments for a given week are to be submitted in class next Wednesday.

Week of March 7 – March 9, 2005

Wednesday, March 9:

Study: Chapters 16 and 17.
Do written assignment (not to be handed in, will not be graded): Chapter 16 exercises 12, 23, 28 (pages 291, 292) and Chapter 17 exercises 4, 14, 16 (Hint: use exercise 15) (pages 307, 308).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 16 exercises 13, 15, 27, 31 (pages 291, 292) and Chapter 17 exercises 5, 15, 25 (pages 307 – 308).

Monday, March 7:

Study: Chapters 15 and 16.
Do written assignment (not to be handed in, will not be graded): Chapter 15 exercises 44, 58 (Hint: use exercise 16 from Chapter 13), 62 (pages 280, 281) and Chapter 16 exercises 4, 8, 10 (pages 290, 291).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 15 exercises 19, 47, 63 (pages 279 – 281) and Chapter 16 exercises 3, 9, 13, 14, 15 (pages 290, 291).

Week of February 28 – March 4, 2005

Friday, March 4:

Study: Chapters 14 and 15.
Do written assignment (due Wednesday, March 9, at the end of the lecture): Chapter 14 exercises 26 (Hint: what is the factor ring?), 54 (pages 261, 263) and Chapter 15 exercises 11, 22 (justify your answer), 40 (Hints: use Fact 1, part 7 from the Handout; where can 1 be mapped to? don't forget that the mapping is assumed to be onto) (pages 278 – 280).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 15 exercises 13, 15, 43, 45 (pages 178, 280).

Wednesday, March 2 and Thursday, March 3:

Study: Chapters 13 and 14.
Do written assignment (due Wednesday, March 9, at the end of the lecture): Chapter 13 exercises 38, 46, 50 (pages 248, 249) and Chapter 14 exercises 10, 30 (Hint: find a larger ideal), 32 (pages 261, 262).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 13 exercises 29, 45, 55 (pages 248, 249) and Chapter 14 exercises 17, 39, 49 (pages 261 – 263).

Monday, February 28:

Study: Chapters 12 and 13.
Do written assignment (due Wednesday, March 9, at the end of the lecture): Chapter 12 exercises 36 (justify your answer), 38, 46 (don't forget to explain why your answer is the smallest subring) (pages 236, 237) and Chapter 13 exercises 16, 30 (pages 247, 248).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 12 exercises 35, 41 (pages 236, 237) and Chapter 13 exercises 9, 11, 13 (page 247).

Week of February 21 – February 25, 2005

Friday, February 25:

Study: Chapter 12.
Do written assignment (due Wednesday, March 2, at the end of the lecture): Chapter 12 exercises 2, 6, 22, 29 (Hint: use one on page A18), 30 (Hint: consider first the case when m≤n) (pages 235, 236).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 12 exercises 3, 21, 31 (pages 235, 236).

Wednesday, February 23:

Study: Chapter 10.
Do written assignment (due Wednesday, March 2, at the end of the lecture): Chapter 10 exercises 10 (Hint: given a symmetry of an n-gon, how can one decide whether it's rotational or reflective?), 12, 20 (Hint: notice that one of the questions is about onto homomorphisms), 34, 42 (Hint: use Fact 4 from the handout), 50 (Hint: H is a kernel of a homomorphism, which one?) (pages 206 – 208).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 10 exercises 7, 13, 33, 53 (pages 206 – 208).

Monday, February 21:

Study: Chapter 9.
Do written assignment (due Wednesday, March 2, at the end of the lecture): Chapter 9 exercises 38, 52, 66 (Hint: you should be able to write down the elements of the subgroup in question explicitly in terms of g and H) (pages 188 – 190).

Week of February 14 – February 18, 2005

Friday, February 18:

Study: Chapter 9.
Do written assignment (due Wednesday, February 23, at the end of the lecture): Chapter 9 exercises 26 (Hint: determine the correct answer by elimination), 34 (Hint: prove first that any such subgroup must contain all squares or real numbers), 36 (Hint: what can |a| be?), 46, 53, 64 (Hint: it's an in-out game), 65 (Hint: see one on page A14) (pages 187 – 190).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 9 exercises 27, 35, 37, 45 (pages 187, 188).

Thursday, February 17:

Study: Chapter 9.
Do written assignment (due Wednesday, February 23, at the end of the lecture): Chapter 9 exercises 4, 8, 10 (Hint: Theorem 4.3 from Chapter 4), 16, 20 (Hint: list all cosets of <(2,2)> first) (pages 186, 187).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 9 exercises 3, 7, 23 (pages 186, 187).

Wednesday, February 16:

Study: Chapter 8.
Do written assignment (due Wednesday, February 23, at the end of the lecture): Chapter 8 exercises 18 (Hint: use Theorem 1 from the handout), 33, 48 (Hint: Aut(Z₂₀)≈U(20)), 56 (Hint: find an isomorphism from U(55) onto U(55)³; what is |U(55)|?) (pages 162 – 164).

Monday, February 14:

Study: Chapter 8.
Do written assignment (due Friday, February 18, at the end of the lecture): Chapter 8 exercises 8, 12, 14 (Hint: use exercise 4), 22, 24 (Hint: consider orders or elements), 42 (pages 162 – 164).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 8 exercises 7, 13, 17, 35, 45 (pages 162 – 164).

Week of February 7 – February 11, 2005

Wednesday, February 9 and Thursday, February 10:

Study: Chapters 7 and 8.
Do written assignment (due Friday, February 18, at the end of the lecture): Chapter 7 exercises 14, 18 (Hint: use problem 4 from the Quiz), 22 (Hint: begin by considering a cyclic subgroup of G), 26, 36 (Hint: if aⁿ=e in G, then what should a be?) (pages 145 – 147).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 7 exercises 15, 17, 25, 29, 37 (pages 145 – 147).

Monday, February 7:

Study: Chapters 6 and 7.
Do written assignment (due Friday, February 18, at the end of the lecture): Chapter 6 exercises 38, 40 (be careful with denominators of rational numbers!) (page 132) and Chapter 7 exercises 6, 8, 12 (justify your answer) (page 145).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 6 exercises 15, 35 (pages 130, 131) and Chapter 7 exercises 3, 5, 11 (page 145).

Week of January 31 – February 4, 2005

Friday, February 4:

Study: Chapter 6.
Do written assignment (due Wednesday, February 9, at the end of the lecture): Chapter 6 exercises 16, 17, 22, 30 (pages 130, 131).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 6 exercises 25, 35, 37 (page 131).

Wednesday, February 2 and Thursday, February 3:

Study: Chapters 5 and 6.
Do written assignment (due Wednesday, February 9, at the end of the lecture): Chapter 5 exercises 6 (Hint: you should be able to present an alement in question explicitely; don't forget to check that it's from A₈), 26 (Hint: use exercise 11), 34 (pages 111, 113) and Chapter 6, exercises 6, 10 (page 130).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 5 exercises 9, 11, 13, 21 (pages 111, 112) and Chapter 6 exercise 27 (page 131).

Monday, January 31:

Study: Chapter 5.
Do written assignment (due Wednesday, February 9, at the end of the lecture): Chapter 5 exercises 4, 17, 28 (Hint: write β in cycle form first), 30, 32 (pages 111 – 113).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 5 exercises 7, 27, 33 (pages 111 – 113).

Week of January 24 – January 28, 2005

Friday, January 28:

Study: Chapter 4.
Do written assignment (due Wednesday, February 2, at the end of the lecture): Chapter 4 exercises 8, 10 (Hint: start by listing all elements of the subgroup in question), 12, 14 (Hint: use the Fundamental Theorem of Cyclic Groups), 50 (Hint: there must be at least 4 more. Why?), 52, 56 (Hint: Fundamental Theorem of Cyclic Groups again) (pages 82 – 85).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 4 exercises 15, 41, 49, 51 (pages 83 – 85).

Thursday, January 27:

Study: Chapter 4.
Do written assignment (due Wednesday, February 2, at the end of the lecture): Chapter 4 exercises 20, 22, 38, 40, 62 (pages 83 – 85).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 4 exercises 11, 21, 27, 31 (pages 82 – 84).

Wednesday, January 26:

Study: Chapter 3 (subgroup tests).
Do written assignment (due Wednesday, February 2, at the end of the lecture): Chapter 3 exercises 2, 8, 44, 54 (ignore the second question) (pages 67 – 71).

Monday, January 24:

Study: Chapter 3 (subgroup tests).
Do written assignment (due Friday, January 28, at the end of the lecture): Chapter 3 exercises 26 (think of what happens if H has an odd element. Hint: odd+even=odd), 34, 42 (don't forget to justify all your assertions), 46 (you don't need to prove that G is a group), 48 (pages 69 – 71).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 3 exercises 33, 43, 47, 49 (pages 70, 71).

Week of January 17 – January 21, 2005

Friday, January 21:

Study: Chapter 3.
Do written assignment (due Friday, January 28, at the end of the lecture): Chapter 3 exercises 4, 6 (Hint: use proof by contradiction), 10 (Hint: you should be able to list elements of the subgroup in question explicitely), 14, 28, 32 for x only (Hint: if |x|=6, what can you say about x¹, x², x³, x⁴, ... Are they all different?) (pages 67 – 70).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 2 exercise 23 (page 55) and Chapter 3 exercises 1, 27, 29, 31 (pages 67 – 70).

Wednesday, January 19:

Study: Chapter 2.
Do written assignment (due Friday, January 28, at the end of the lecture): Chapter 2 exercises 19 (Hint: use induction. Don't forget the case n<0!), 26, 34, 35 (pages 55, 56).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 2 exercises 21, 25, 33 (pages 55, 56).

Week of January 10 – January 14, 2005

Friday, January 14:

Study: Chapters 1 and 2.
Do written assignment (due Wednesday, January 19, at the end of the lecture): Chapter 1 exercises 16 (justify your answer), 22 (have a look at Figure 1.5 on page 37) (pages 38, 39); Chapter 2 exercises 6 (Hint: don't go further than D₄), 8, 14 (Hint: find such c (in terms of a and b) that the equation ab=ca is obviously true. What can one conclude from b=c then?), 18, 17 (Hint: take inverse of both parts of the equation and use exercise 18. What is the inverse of ab, by the way?) (pages 54, 55).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 2 exercises 1, 3, 13, 15 (pages 53, 54).

Thursday, January 13:

Study: Chapter 1.
Do written assignment (due Wednesday, January 19, at the end of the lecture): Chapter 1 exercises 2, 3 (justify your answer), 6 (Hint: recall what determines position of an n-gon geometrically), 8, 10 (use exercises 6–8), 14 (pages 37, 38).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 1 exercises 7, 9, 15 (pages 37, 38).

Wednesday, January 12:

Study: Chapter 0 (Equivalence Relations).
Do written assignment (due Wednesday, January 19, at the end of the lecture): Chapter 0 exercises 46, 48 (page 25).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 0 exercise 47 (page 25).

Monday, January 10:

Study: Chapter 0 (Mathematical Induction).
Do written assignment (due Friday, January 14, at the end of the lecture): Chapter 0 exercises 22, 26, 27 (Hint: use exercise 9), 28 (Hint: don't try to use induction here, it won't help. Why?), 42 (page 24).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 0 exercises 20, 21 (page 24).

Week of January 5 – January 7, 2005

Friday, January 7:

Study: Chapter 0 (Modular Arithmetic).
Do written assignment (due Friday, January 14, at the end of the lecture): Chapter 0 exercises 4, 8 (Hint: use exercise 7 without proving it), 9, 14, 16, 17 (Hint: use the Fundamental Theorem of Arithmetic) (pages 23, 24).
Problems to practice (do not hand in solutions to these problems, they will not be graded): Chapter 0 exercises 7, 12, 15 (Hint: use exercise 9) (pages 23, 24).

Wednesday, January 5:

Study: Chapter 0
Do written assignment (due Friday, January 14, at the end of the lecture): Chapter 0 exercises 6, 10 (Hint: use proof by contradiction), 18, 19 (pages 23, 24).