Math 441 Hints

Hints for students taking Math 441, "Introduction to Modern Algebra," using the text Numbers and Symmetry, by Johnston and Richman

See Fred Richman's web site at http://www.math.fau.edu/Richman/

See Errata for textbook.

Chapter 13

• For #3 in Chapter 13, the same general approach used in Problem 2 should work. You will want to use the expression n(n+1)/2 for the sum of 1 through n. Try working from both ends to see if you can reach the same point in the middle. (Afterward, of course, you want to rewrite it so it goes steadily from one end to the other.)
  
  
• In #6, try this approach for "guessing" the formula: look at successive differences in the terms, and see how many times you need to take differences before you get a constant sequence.

  Example:  The sum of the first n squares:    1          5        14        30        55        91
  Differences between successive terms:               4        9          16        25        36
  Second iteration:                                         5        7           9          11
  Third iteration:                                                2         2           2
  

  After three iterations, we have constants. This tells us that the sum of the first n squares can be written as a degree-3 polynomial in n. So we write:

  S(n) = a n³ + b n² + c n + d,

  And substitute numbers for n to get 4 equations involving the unknown coefficients, which we then solve. I suspect that the sum of 4th powers may be a degree-5 polynomial, but you should check by the method of differences.

  Of course, this is only the "guessing" part. You still need to prove by induction that the formula you've got will work for all n.

Chapter 1

• Section 1.1 #3. Here you need to compare complex numbers z with the corresponding numbers z*(2+3i). Represent both the original z and the product as vectors from the origin. We are seeking a geometric description not of either vector itself, but of the transformation required to turn z into z*(2+3i). How much do you need to rotate z, and by what factor is its length multiplied, in this transformation?
  
  
• Section 1.1 #5. Here again, we want to think about the geometric transformation that happens when you multiply numbers by (1 + i), (1 - i), and 2. We want an understanding of why doing the first two transformations has the same ultimate effect as simply multiplying by 2.
  
  
• Section 1.2 #12. Follow the Prime Directive for Writing Proofs: start with the definitions! We know that ab is a unit. What precisely does this mean? We seek to show that a is a unit. What exactly does that mean?

Chapter 3

• Bezout's Equation and Relative Primality. Note that Johnson and Richman define natural numbers a and b to be relatively prime if sa + bt = 1 for some integers s, t. This says nothing about common factors, or prime factorization. This is the definition you should use in the exercises. J & R choose this definition because of their constructivist approach: they prefer to give a positive definition (there are s and t such that sa + bt = 1) rather than a negative one (there is no number d > 1 which divides both a and b). J & R's definition has some advantages: some proofs are easier this way (others are harder), and this definition makes sense even in situations where a number may not have a unique factorization into primes.



• In Problem 9 on page 45, J & R refer to the proof of the Euclidean algorithm for the Gaussian integers. This is implicit in the discussion on pages 43 and 44. What needs to be done here is:
  • The algorithm needs to be described
  • You need to prove that the algorithm will end
  • You need to show that when the algorithm ends, we have a GCD

Chapter 6

• Once the square has been moved, there are two potential meanings for each letter: the original corner with that label, or the current position of that label. By convention, we stick with the original position.

  To see clearly what this means, cut out a square of paper, and label its corners A, B, C and D (clockwise, starting from the upper left). Also trace around the square on your desktop, and label the four corners of the fixed square with the same letters. Now a 90-degree clockwise rotation is represented by the notation BCDA, meaning A->B, B->C, etc. If we follow this rotation by the transformation ADCB, this means that we flip about the diagonal from upper left to lower right -- that is, we interpret ADCB in terms of the labels on the desktop, not the labels on the square. See the sketch below.


• Note that the inverse is always the operation that returns the square to its original position. The following table may help.

  Symmetry		Inverse
  Block Notation	Cycle Notation	Block Notation	Cycle Notation
  BCDA	(ABCD)	DABC	(ADCB)
  ADCB	(BD)	ADCB	(BD)
  BADC	(AB)(CD)	BADC	(AB)(CD)

• The cycle notation is preferred for writing permutations. In section 6.3 and beyond, please use cycle notation. The transformations in the sketch above are represented by cycle notations (ABCD), (BD), and (AD)(BC).




Dihedral Group Explorer

Here is a tool for experimenting with small dihedral groups.

You might use this tool in exploring the following questions:
• What are the different cycle shapes possible in Dihedral groups of various orders? (The "cycle shape" of an element is just how many cycles of what lengths it has. For instance, three cycles of length 2.)
• What are the different orders possible for elements of a Dihedral group? (See order of an element below.)
• Can you tell from the cycle shape of an element what the cycle shape of its inverse will be?
• Can you tell from the cycle shape of an element what its order will be?
• Can you tell from the order of an element what its cycle shape is?
• Is it true that a flip followed by a rotation is its own inverse?
• What do you get if you sandwich a clockwise rotation between two flips?
• What do you get if you sandwich a flip between two clockwise rotations? Two counterclockwise rotations? One of each?
• Can you combine the available flips and rotations to get any flip you want? E.g., in D₆, how do you get the transformation (2 6)(3 5)? How about (1 4)(2 3)(5 6)? Can you come up with a general rule for how to get these transformations?



• Definition of Order. Exercise 9 on p. 93 involves the order of a group element, which is not actually defined until the bottom of p. 98: the order of an element r is the smallest positive integer n for which rⁿ is the identity transformation.

Permutation Tool

Here is a tool for doing some simple calculations with permutations. Write any permutation as a product of cycles, and this tool will reduce it to a "canonical" form, as a product of disjoint cycles. Remember to separate elements with commas or spaces. If you write (123), it will be interpreted as the permutation that sends the element 123 to itself.

A canonical form means a form that is unique for each permutation. There are many ways to write any particular permutation: these all represent the same permutation:

• (1 2)(3 4)
• (1 2)(4 3)
• (4 3)(1 2)
• (1 2)(5)(3 4)
• et cetera

1. Use cycle form
2. Write as a product of disjoint cycles
3. Omit "cycles" of length 1, except that the identity permutation is written as (1)
4. Rotate each cycle to put its smallest element at the left
5. Order the cycles from left to right by smallest element

There is still one kind of ambiguity: there is no way to tell whether a permutation such as (1 2 3) is an element of S₅ or, for instance, S₁₇. This is not usually a problem, since the context usually resolves the question.

Chapter 8

• p. 132 The definition of index is not clearly stated. The index of H in G is the number of left cosets of H in G.

Chapter 9

• p. 152 contains an error. The symmetry group of a rectangle does act transitively on the vertices of the rectangle.



• #7 on p. 160: Rather than looking at all the squares, it should be sufficient to look at all the possible cycle shapes. E.g., (a b)(c d e) is one of a handful of cycle shapes that can occur in S₅. What are all the cycle shapes in S₅? What is the square of each? If we multiply together two squares, what are the possible cycle shapes? E.g., in S₅, 3-cycles could overlap in 1, 2 or 3 elements. If overlap exceeds 1 element, there are two different ways the overlap could work. Consider (1 2 3)(2 3 4) vs (1 2 3)( 3 2 4). )

  As Exercise 8 shows, no general argument is possible, since the result is no longer true once n exceeds 5.




• p. 161

  There is an error in the second full paragraph. The symmetry group of the second frieze pattern is {1, m_y}. The notation m_y means a reflection that keeps every point on the y-axis fixed, and flips things left-for-right. (Thanks to Cathy Tousey for tracking down J&R's several incompatible uses of the notation, and figuring out which one was the error.)

  The symmetry group of frieze pattern #2 consists of

  • (1 | n ), for all integers n. These are translations by n units to the right.
  • (m_y | n ), for all integers n. These are left-to-right reflections, followed by translation by n units to the right.
  
  
  Remember that the symmetry on the right acts first. So, for instance,

  ( m_y | 4 )( 1 | 3 ) = ( m_y | 1 )

  Here we have:
  • a translation 3 units to the right, followed by...
  • a left-to-right reflection, followed by a translation 4 units to the right
  
  
  The net effect is the same as a left-to-right reflection followed by a translation 1 unit to the right.

Errata for Textbook

(This list is from Fred Richman's web site. At least some of these errors were corrected by the time my copy of the text was printed.)

• Page 5, paragraph 3: (3+5i)+(6-2i) = 9+3i, not 9-3i.
• Page 5, paragraph 3: (3+5i)(6-2i) = 18+24i-10i², not 18+24i+10i². This error is repeated in paragraph 4, and in the first display on page 6.
• Page 28, Exercise 1, N(rho) and N(alpha) should be |N(rho)| and |N(alpha)|.
• Page 29, last line of second display: 21X³- 17X² + 9X - 1, not 21X³- 17X² + 5X - 1.
• Page 42, just before the exercises: should be t - 1 = -n4, not t - 1 = n4, and t = 1 - 4n, not t = 4n - 1.
• Page 214, line 21: should be uG, not vG.