\( \newcommand{\lb}{\langle} \newcommand{\rb}{\rangle} \newcommand{\V}{{\sf V}} \newcommand{\W}{{\sf W}} \newcommand{\bfr}{{\bf R}} \newcommand{\bfz}{{\bf Z}} \newcommand{\span}{{\rm span}} \newcommand{\T}{{\sf T}} \newcommand{\mnn}{M_{m\times n}(\bfr)} \newcommand{\a}{\alpha} \newcommand{\b}{\beta} \newcommand{\g}{\gamma} \newcommand{\d}{\delta} \renewcommand{\l}{\lambda} \newcommand{\abcd}{\left( \begin{array}{rr} a&b\\ c&d \end{array}\right) \newcommand{\va}{{\bf a}} \newcommand{\lb}{\langle} \newcommand{\rb}{\rangle} \newcommand{\mod}{{\rm mod}} } \)

Homework Rules and Assignments
MAS 4301 Section 075A (14554) — Abstract Algebra 1
Spring 2024

Last updated   Wed   May 1   2024   02:00  EDT

  • General information
  • Academic Honesty
  • Homework Rules
  • Assignments

    General information

    Academic honesty

    Some Rules for Hand-In Homework

        Even when homework is well written, reading and grading it is very time-consuming and physically difficult for your instructor. In order that this process not be unnecessarily burdensome:

    • Assignments

    Unless otherwise indicated, problems are from Gallian. A problem listed as (say) "Ch. 8/ 17" means exercise 17 at the end of Chapter 8.

    Date due Assignment
    F 1/12/24 Assignment 0 (just reading, but important to do before the end of Drop/Add)

  • Read the Class home page, and Syllabus and course information handouts.

  • Read all the earlier information on this page (except that you may skip the paragraph about LaTeX; that's optional reading)

  • Go to the Miscellaneous Handouts page and read the handouts "What is a proof?" and "Mathematical grammar and correct use of terminology". (Although this course's prerequisites are supposed to cover most of this material, most students still enter subsequent courses without having had sufficient feedback on their work to eliminate common mistakes or bad habits.)
        I recommend also reading the handout "Taking and Using Notes in a College Math Class," even though it is aimed at students in Calculus 1-2-3 and Elementary Differential Equations.

  • In the book by Hammack that's the first item on the Miscellaneous Handouts page, read Section 5.3 (Mathematical Writing), which has some overlap with my "What is a proof?" and "Mathematical grammar and correct use of terminology". Hammack has a nice list of 12 important guidelines that you should already be following, having completed MHF3202. However, most students entering their subsequent courses violate almost all of these guidelines. Be actively thinking when you read these guidelines, and be ready to incorporate them into your writing. Expect to be penalized for poor writing otherwise.
        I'd like to amplify guideline 9, "Watch out for 'it'." You should watch out for any pronoun, although "it" is the one that most commonly causes trouble. Any time you use a pronoun, make sure that it has an clear and unambiguous antecedent. (The antecedent of a pronoun is the noun that the pronoun stands for. It needs to appear either in the same sentence as the pronoun, or in the immediately preceding sentence.)

  • Read Gallian Chapter 0, with the following exceptions:

    • Replace the first definition on p. 21 with the portion of my Sets and Functions handout from the bottom of p. 6 through the material on the word "image" on p. 8. (I corrected some typos in my 2024 version of this handout on 1/10/24. See the current version if there was anything that seemed weird or confusing when you read the version I'd previously posted.)
          I think you will also find it useful to read the rest of this handout, but that's not urgent. Note: I originally wrote this handout for students who hadn't taken MHF3202 (at a time when MHF3202 wasn't yet a prerequisite for anything), so the level may initially seem very elementary. But don't be fooled: these notes include some material that most students who've completed MHF3202 (or even MAS4105) are, at best, unclear about, especially when it comes to writing mathematics.

    • In the "Modular Arithmetic" section, feel free to skip everything from the beginning up Example 4 up through about the middle of p. 12 (the sentence "We have successfully corrected the error"). Optionally, if you're interested in the topic of those examples, feel free to read those pages when you're done with the rest of this assignment.

    • It's okay to skip the section on Complex Numbers for now. (I may have you read it later.)
    • W 1/24/24
    		 Assignment 1

  • Read the "Inductive proofs ..." handout posted on the Miscellaneous Handouts page.

  • Do these non-book problems. (Some of these aren't really "non-book" problems; they're "proofs left to the reader" from Gallian, or slight modifications of these, that just aren't among his numbered exercises.)

  • Ch. 0/ 57, 65

  • Read Chapter 1.
        Note: What I called a "multiplication table" in class is what the book calls an operation table or Cayley table.
        Note: It is is true that the property referred to as "closure" on p. 33 is indeed called closure, and the terminology is perfectly reasonable in the context of this chapter. It is also true, in the more general setting of the next chapter (which will be added to this assignment after the next lecture), that this property is generally called closure (see p. 42). But in the context on p. 42, "closure" is somewhat of a misnomer; the definition of "binary operation" on p. 42 makes the term "closure" unnecessary and a bit wrong-headed; note that it doesn't even appear in the definition of "group" on the next page. See my comment after "Read Chapter 2" below.

  • Ch. 1/ 1, 2, 4, 5, 16 (recall that a rhombus is a parallelogram all four of whose sides have equal length).
        Regarding #5: (1) The \(n=3\) and \(n=5\) cases (exercises 2 and 4) should give you a jumping-off point for the general odd-\(n\) case. To get the general even-\(n\) case, I suggest you first figure out the case \(n=6\), to give you a second example to accompany the \(n=4\) case that you've already seen. (2) In this problem, you're not being asked to prove that your answer is correct. The first part of the problem just asks you to describe (all) the elements of \(D_n\), analogously to the way they were described for \(D_4\) in Chapter 1. But you should show how your description leads to your answer to "How many elements does \(D_n\) have?"

  • Read Chapter 2.
        Regarding "closure" (and the related adjective "closed"):
      Given a set \(S\), a subset \(H\subseteq S\), and binary operation, say \(\star\), on \(S\), a question that is often very important is this: Given \(a,b\) in the subset H, does \(a\star b\) always lie in \(H\)? If so, we say that \(H\) is closed under \(\star\).

      The question of whether a given subset \(H\) of \(S\) is closed under a given binary operation on S is not interesting unless \(H\) is a proper subset of \(S\) (a subset that is not all of \(S\)). The reason is that by definition, all outputs of a binary operation on \(S\) lie in \(S\); there's no larger universe in which the outputs could potentially lie. By the definition of "binary operation" on Gallian p. 42 (which is correct and current), and his definition of closure on p. 42, every binary operation is closed. The term "closed binary operation" is redundant.

      Thus, for my money, the p. 43 sentence in Gallian after the definition of "group" (in particular, the phrase "any pair of elements can be combined without going outside the set") is misleading; it fosters a muddled understanding of the definition of "binary operation" on the previous page. In this (and any modern) definition of binary operation on a set \(S\), "going outside the set" is meaningless. For students (or anyone else) who know what the definition of a function is, there is no need for an extra word like "closure" to tell us that for a function from \(S\times S\) to \(S\), the outputs all lie in \(S\). Indeed, introducing this extra word detracts from the definition of "function from A to B."

      The use of "closure" on p. 42 of Gallian dates back to an era before the accepted definition of "\(\star\) is a binary operation on \(S\)" evolved (long ago!) to one that required \(a\star b\) to defined for all elements \(a,b\in S\). In that era (e.g. in the well respected and often-cited 1959 textbook The Theory of Groups, by Marshall Hall), a binary operation on \(S\) was something that is now called a partial binary operation: a function from any subset of \(S\times S\) to \(S\).

      However, the p. 43 sentence after Gallian's definition of "group" attaches a meaning to "closure" that is different from Hall's (the latter being closer to the meaning on p. 42). In p. 43's usage of "closure", what determines whether a binary operation on \(S\) is closed is whether the codomain of the operation is \(S\), not whether its domain is all of \(S\times S\). From this sentence, and the first several exercises at the end of Chapter 2, we can infer that what Gallian means by "binary operation" is not what he defined on p. 42. Rather, what he apparently means by "binary operation on a set \(S\)" is "function from \(S\times S\) to any set \(T\)"; he calls the operation closed if \(T=S\) (or, more generally, if \(T\subseteq S\)). Gallian may not be the only author who does this, which is why I'm not branding it as outright wrong; I'm only pointing out that this usage of closure/closed in combination with binary operation is inconsistent with the the definitions on p. 42, and also inconsistent with the (now defunct) usage in Hall's book.

      Although he doesn't say so explicitly, Gallian's usage of closure/closed (but not binary operation) from p. 43 onward is essentially the one I gave at the start of this comment. In Chapter 2's exercises 1 and 4, observe that in each case, what Gallian means to ask are being asked is (implicitly) a question of the form in my first paragraph above: the data you're given are (i) a set \(S\) in the background (one that Gallian never mentions explicitly in any part of exercise 1), (ii) a true binary operation \(\star\) on that unnamed background set \(S\), and (iii) an explicitly stated, proper subset \(H\subsetneq S\). What you are asked to check is whether H is closed under that operation \(\star\) on \(S\). For example in 1a, \(S={\bf Z}\) and \(H={\bf Z}_+\); in 1b, \(S={\bf Q}\) and \(H={\bf Z}\setminus\{0\}\); in (1c), \(S\) is the set of all functions from \(\bfr\) to \(\bfr\).

  • (a) Find correct wording for Ch. 2 exercise 1 that expresses the question that Gallian meant to ask. (As noted above, by Gallian's own definitions, every binary operation is closed, so the question as written makes no sense. Correct wording should involve a phrase like "such-and-such set is/isn't closed under such-and-such binary operation on such-and-such [larger] set.") If you need different wording for each part, that's fine. (b) Do the reworded exercise. In part 1e, replace "integers" by "nonzero integers". Exponentiation of nonzero integers means the function \( (m,n)\mapsto m^n \), where \(m\) and \(n\) are nonzero integers. The "nonzero" restriction avoids the problem that \(0^n\) is not defined when \(n\leq 0\), but does not change the answer to this problem-part. (Without the "nonzero" restriction, there are two reasons why the answer is what it is; with the restriction, there's only one reason.)

  • Ch. 2/ 2–4, 8, 9, 11–15, 23, 24, 26–28, 33, 34 (the first half is enough), 47, 49.
    Also do this exercise that's almost the same as #26: Show that for two elements \(a,b\) in a group, if \(b=a^{-1}\) then \(a=b^{-1}\).
    (Although this looks like a long list of exercises, you should find that most of them are very short!)

    On Wednesday 1/24, hand in only the following problems:

    • Non-book problem NB 1.3
    • Ch. 1/ 16
    • Ch. 2/ 8, 12, 14, 26, 28, 34 (just the first half) Note: Unless I say otherwise, the only proofs you should be handing in for homework are for exercises that ask you to prove or show something.

      Some reminders about proofs:

      1. Don't make statements involving objects (groups, elements, etc.) for which you haven't introduced notation, unless that notation was fixed by the wording of the hypotheses (or problem-statement) you were given.

      2. Notation introduced in an "if" sentence or clause is extremely short-lived; it's "dummy" notation that does not survive past the end of the sentence. This is true whether the "if" statement is in a problem-statement, in the hypotheses of a theorem, or in the body of your proof. For example, if a problem-statement says "Show that if \(G\) is a group, then ...", the letter \(G\) has not been assigned any meaning that you can carry over into your proof. You would need to start your proof with a sentence (or phrase) like "Let \(G\) be a group," or "Let \(H\) be a group," or "Assume [or Suppose] that \(K\) is a group." (The letter you choose could be \(G\) or anything else that doesn't already have an assigned meaning in the context of the theorem or problem. The word "suppose" is an alternative to "assume".)

      3. Notation that's properly introduced by a "let" or "assume" phrase does survive, all the way up through the end of the proof (or argument within a proof, if the notation was introduced just for that argument). This is true, again, whether the "if" statement is in a problem-statement, in the hypotheses of a theorem, or in the body of your proof. But once notation has been introduced by a "let" or "assume" phrase, that notation is fixed for the rest of the proof (or argument); you can't just "let" the same notation have a new meaning later in the proof. And when notation is fixed by the hypotheses (or problem-statement), you shouldn't re-introduce that notation in your proof. For example, if the hypotheses include "Let \(G\) be a group," you don't have the power later to "let" \(G\) be what it already is.

      4. Statements with "for all" quantifiers do not introduce surviving notation for the object being quantified. For example, a sentence like "\(f(x)>0\) for all \(x>1\)" does not leave you with anything called \(x\) that you can make a statement about; a follow-up statement of the form "Since \(x>1\), such-and-such is true" would make no sense.

      5. A hint for a problem is not part of the problem-statement, so notation introduced in a hint never survives past the end of the hint. (This applies to "suggestions" as well, since a "suggestion" is just a type of hint.)
    • W 2/7/24

    (so that you have time to understand all my comments on previous assignment, and be careful not to make the same mistakes)

    		Assignment 2

  • Read Chapter 3.

  • Ch. 3/ 1, 2, 4, 5, 7, 9–13, 18, 19, 21, 28, 32, 34, 36. Several of these problems occur in pairs, with the first problem helping you do the second problem more easily if you notice that the problems are related. (Note that "more easily" does not necessarily mean easily.) For example, 4&5 and 9&10 are two such pairs. Some suggestions:
    • If you're stuck on #32, you may find it helpful to do NB 2.3 first. However, #32 can be done without ever doing NB 2.3.

    • For #34, I suggest using the definition of "subgroup" I gave in class. (Or: use the book's definition, but also use the "Two-Step Subgroup Test". Recall that what I did in class amounted to taking the "Two-Step Subgroup Test" as the definition of "subgroup", and deriving the relation between \(H\) and \(G\) in the book's definition of "subgroup" as a consequence, rather than the other way around.)

  • Do these non-book problems.

  • Read the file "lecture_notes_2024-01-22.pdf" posted on Canvas under Files. This file gives a clean version of a garbled proof I gave in class, but covers some additional material as well.

  • Reminder: EVERY handout that I assign you to read, and EVERY rule, comment, guideline, etc. that I insert into these assignments, is MANDATORY reading for EVERY student. I expect students to read these carefully—not just to give them (at most) a quick glance, saying to themselves, "I've seen this before, and got A's in all my previous classes. I don't need to pay careful attention to this (or read it at all)." Many of my handouts (etc.) cover common mistakes that I see all the time, even in work by students whose previous teachers told them that their writing is great. There aren't enough hours in the week for any instructor to correct the same, common mistakes on a significant number (sometimes almost all) students' papers (and many instructors deal with this by simply not correcting these mistakes.) When you enroll in college, there's an implicit understanding that you're going to be taught in a group; that your teachers won't have time to make any and all corrections in private communications with each student.

    Promptly reading all comments I make on your work is also mandatory. Once I return graded work to you, if I've made a comment or correction you don't understand (whether because of content or illegibility), ask me for clarification it in my next office hour that you can make. After I've made a correction, I should not see the same issue in your future work (or at least should see it only rarely).

  • Read Chapter 4.

  • Ch. 4/ 1, 2, 4, 7, 8, 9, 16, 19, 31, 39, 52. See notes about some of these problems below.
    • #7: You've already seen an example of such a group, as recently as in the Chapter 3 exercises above. But if you have trouble with this, one of the non-book problems should help you find another example (one you've also already encountered, though you may not have been conscious at the time that it has the properties we're looking for in Ch. 4 #7).

    • #8: In each of parts (a), (b), and (c), you should find that all the listed elements have the same order.

    • #16: Assume \(a\) has finite order.

    • #39 has a typo: "\(|ab|\)  has infinite order" should have been either "\(ab\)  has infinite order" or "\(|ab|\)  is infinite."

    On Wednesday 2/7, hand in only the following problems:

    • NB 2.3, 2.4
    • Ch. 3/ 2, 10, 18, 28, 32, 34 (first sentence only), 36

    • Ch. 4/ 4, 8, 16, 52

    Note: proofs for most of the book problems are very straightforward, so a lot of what I'm grading you on is how well you write your proofs.

    • M 2/19/24
    (no hand-in problems)    		 Assignment 3

  • Read Chapter 5.

  • Read the handout "Gallian_lemma_chap5-corrected_proof.pdf" posted on Canvas under Files.

  • Ch. 5 / 3–7, 9, 15, 14 (I recommend doing 15 before 14), 19, 23, 25, 30, 34 (Note: distinct is not the same as disjoint), 41, 43, 44, 49, 58

  • Read Chapter 6 up through Example 5.

    No homework will be collected for Assignment 3.

    • M 2/19/24

    First midterm exam

          The cutoff for the "fair game" material for this exam is the end of Chapter 5.

    new due-date:
    W 3/6/24     		Assignment 4. I have changed the due-date to Wednesday Mar. 6. I'm working on the list of hand-in problems now, and should have it posted soon.

  • Read the handout Difference Between Inverse Functions and Inverse Images .

  • Read (all of) Chapter 6.
      Comments on Chapter 6:

    • In the proof of Theorem 6.5, I think the reference to "Example 13" on the fourth line was supposed to be another reference to Example 15. The current Example 15 was probably Example 13 in some older edition of the book.

    • Do not continue until you have read the "Difference Between Inverse Functions and Inverse Images" handout.

    • Do not continue reading these comments until you have read the "Difference Between Inverse Functions and Inverse Images" handout.

      Theorem 6.3, part 5, needs some elaboration; the presentation is a bit sneaky. In the statement of property 5, observe that \(\phi^{-1}(\overline{K})\) is defined as the inverse image of \(\overline{K}\) under \(\phi\), not as the image of \(\overline{K}\) under \(\phi^{-1}\). The proof of property 5, however, is written as if \(\phi^{-1}(\overline{K})\) had been defined the latter way, i.e. as \(\{\phi^{-1}(x): x\in \overline{K}\}\).

      Property 5 does not, in fact, follow directly from properties 1 and 4; you need to know that when a function \(f:X\to Y\) has an inverse function, the inverse image of any \(U\subseteq Y\) under \(f\) coincides with the image of \(U\) under \(f^{-1}\).

      A good question to ask is: if the author wanted to prove property 5 the way he indicated, why didn't he simply define \(\phi^{-1}(\overline{K})\) as the image of \(\overline{K}\) under \(\phi^{-1}\)? Why bother defining \(\phi^{-1}(\overline{K})\) instead as an inverse image, then relying on an unstated equivalence that students are likely either to overlook or to have difficulty appreciating? (Any instructor who has ever asked students to hand in proofs involving inverse images knows that students have difficulty with the concept, especially the difference between inverse images and inverse functions. Students generally don't become aware of this difficulty until they hand in exercises on this topic, and receive feedback.)

      I'm pretty sure the reason is that Gallian wanted the wording of property 5 in Theorem 6.3 to be exactly the same as the wording he uses later for property 7 of Theorem 10.2, which we'll get to in a few weeks. (He could have decided this when revising an older edition, and changed the statement of Theorem 6.3 without making corresponding changes to the proof.) Theorem 10.2 is about homomorphisms, a generalization of isomorphisms. A homomorphism \(\phi\) doesn't have to be one-to-one or onto, so doesn't have to have an inverse function (in fact, the only homomorphisms that have inverse functions are isomorphisms); in Theorem 10.2, property 7 has to be stated in terms of inverse images under \(\phi\), since it can't be stated in terms of images under a (generally nonexistent) inverse function \(\phi^{-1}\).

    • Ch. 5/ 35. (Added Wednesday, but should have been added earlier. The result of this exercise is needed before Ch. 6/ 21 can even make sense.)

        Just FYI: when I was writing the exam, I originally was going to use this problem in place of problem 3. All "Show [this thing] is a subgroup" problems are essentially the same, and I thought that Ch. 5/ 35 might be a few lines shorter to write out, but it would have been one more problem involving the symmetric group \(S_n\).

    • Do these non-book problems.

    • Ch. 6/ 2, 4, 5, 7, 8, 9, 11, 12, 21, 22, 24, 29 (as modified below), 30–32, 34 (as modified below), 41, 42.

        Notes on some of these problems:

      • #21. The non-book problems will help with this one; it's essentially a special case of NB 4.2(c). For #21, the sketch in the back of the book is only a sketch of a proof, though it may have an appearance of being a proof. The sketch omits the first step, in Gallian's own itemized four-step procedure, needed to show that two groups are isomorphic.

      • #29. Modify this exercise as follows: (a) For each integer \(r\) relatively prime to \(n\), show that the map \(\a_r: \bfz_n\to\bfz_n\) defined by by \(\a_r(s)=rs\mod n\) is an automorphism of \(\bfz_n\). (b) Show that if \(r_1,r_2\in \bfz\) differ by a multiple of \(n\), then \(\a_{r_1}=\a_{r_2}\). (Recall that two functions \(f,g\) are equal iff \(f\) and \(g\) have the same domain \(D\), have the same codomain, and \(f(x)\)=\(g(x)\) for every \(x\in D\).) Hence, for each \(r\in \bfz\), the automorphism \(\a_r\) is the same as the automorphism \(\a_{r\, {\rm mod}\, n}\).

      • #32. Hint. As shown in the class , the automorphisms of \(\bfz_n\) are all of the form in #29.

      • #34. Modify this one by adding to the end of property 4 in Theorem 6.3: "... and \(\phi\) carries \(K\) isomorphically to \(\phi(K)\)." (This means that \(\phi\), with a restricted domain of \(K\) and a restricted codomain of \(\phi(K)\), is an isomorphism from \(K\) to \(\phi(K)\). Note that "\(\phi\) with a restricted domain of \(K\) and a restricted codomain of \(\phi(K)\)" is not literally the same function as \(\phi\); for two functions \(f\) and \(g\) to be the same, they have to have the same domain, say \(A\) same codomain, say \(B\), and satisfy \(f(x)=g(x)\) for all \(x\in A\). We could reflect the fact that the [doubly] restricted function above is not literally the same as \(\phi\) by choosing a different letter, or by "decorating" the originaly letter \(\phi\) appropriately; e.g. we could call the new function \(\phi|_K^{\phi(K)}\). The latter notation would be so clumsy and distracting that I am avoiding it here. The simpler "\(\phi|_K\)" would not work, because this simpler notation already has a standard meaning: it's the function obtained from \(\phi\) by restricting only the domain [in the setting of Theorem 6.3, \(\phi|_K\) would be a function from \(K\) to \(G\), not from \(K\) to \(\phi(K)\)].)

    • Re-read the green "Some reminders about proofs" under Assignment 1. Wherever I wrote a comment on your exam that said something like "See HW1," figure out which of the reminders I was re-reminding you of.

            Similarly, if I wrote any comment on your exam that referred to "HW0", go through all the Assignment 0 readings until you find what specifically I was referring to. And when doing any reading I assign, make sure you follow my emphatic "Be actively thinking ..." instruction from Assignment 0.

            Several weeks ago, when reading Chapter 3, I hope you noticed and followed some similar advice Gallian gave on p. 46: "With the examples given thus far as a guide, it is wise for the reader to pause here and think of his or her own examples. Study actively! Don't just read along and be spoon-fed by the book."


          Whether you've never gotten this kind of advice before, or have gotten it till you're blue in the face, it's still good advice. Not reading (or skipping over portions of assigned reading), and not reading actively, are among the biggest obstacles to doing well in any conceptually focused class.

    • Also re-read the blue reminder in Assignment 2. You should regard doing 100% of assigned reading, and seriously attempting 100% of assigned exercises (not just hand-in exercises) as the bare minimum needed for a grade of C. "C" means "satisfactory". Failing to do required work is not satisfactory.

    • Read the posted solutions to the first exam.

      On Wednesday 3/6, hand in only the following problems:

      • NB 4.1. (Note that what you, the student, have to do is just what's in the last sentence. Everything up to that point was there just to show how you might discover, on your own, the formula I gave for \(\phi\); it wasn't pulled out of thin air. None of that discussion/motivation should appear in your proof.)

      • Ch. 6/ 4, 12, 32, 34 (as modified above), 42.
    • F 3/29/24
    (no hand-in problems)    		 Assignment 5

  • Read Chapter 7 up through Theorem 7.5.

  • Ch. 7/ 2, 3, 6, 8, 9, 11, 13, 14, 15, 17, 18, 19, 25, 33, 49
      Notes on some of the Chapter 7 problems:

    • #8 has a bad typo! The definition of \(HK\) should be the same as in Theorem 7.2, namely \(HK=\{hk\mid h\in H, \ k\in K\}\). (Thanks go to Cameron for bringing this to my attention.)

    • #13. (1) In the second sentence, "If \(\bfr^+\subseteq H \subseteq \bfr^*\)" could be replaced simply by "If \(H\) contains \(\bfr^+\)", since \(H\) was already assumed to be a subset of \(\bfr^*\). (2) Add a "part (b)" to this exercise: What is the index of \(\bfr^+\) in \(\bfr^*\)?

    • #14. First show that \(H\) is a group. (If you need to review complex numbers, see Chapter 0.) This set \(H\) is called the unit circle (in \({\bf C}\)).

    • #19. Fermat's Little Theorem (Corollary 5 in this chapter) is very helpful here!

  • Do these non-book problems (of which there are now three).

  • Read Chapter 8 up through, but not including, the sentence near the bottom of p. 161 beginning with "With this goal in mind ... ." The first sentence of the section "The Group of Units Modulo \(n\) as an External Direct Product" (p. 160) gives good motivation for this section, but even I found much of the presentation hard to follow:

    • The first sentence of Theorem 8.3 has a clear relation to the topic of this chapter, but the second sentence does not. It took me a while to figure out why the second sentence—which has nothing to do with direct products—is stated in the same theorem. No insight into the purpose of the second sentence is provided in this chapter.

    • The "proof" on p. 161 is just a sketch of a proof. Some of the verifications are far from obvious, and it's unreasonable to expect many students to figure out how to do them. Even if you've done Exercise 17 of Chapter 3 and Exercises 9, 17, and 19 of Chapter 0—none of which I assigned, but none of which is difficult—you should be able to show, without too much difficulty, that the relevant maps in this argument are operation-preserving and one-to-one. It's less obvious how to use those exercises to that the indicated maps from \(U_s(st)\) to \(U(t)\)are onto, and (as far as I can tell) those exercises don't leave you even with a plan of attack for showing that the indicated map from \(U(st)\) to \(U(s)\oplus U(t)\) is onto.

      What's going on in Theorem 8.3 is that the relation of \(U(st)\) to the subgroups \(U_t(st)\) and \(U_s(st)\) is that \(U(st)\) is an excellent example of an internal direct product (of these two subgroups), something that's not defined for another 20+ pages in Chapter 9. (Indeed, this very example is given as Example 17 of Chapter 9, though even there none of the argument that was missing in the "proof" of Theorem 8.3 is supplied. I may write up a proof.) This, together with the fact that \(U_t(st)\approx U(s)\) and \(U_s(st)\approx U(t)\), is what motivates bothering to mention the groups \(U_t(st)\) and \(U_s(st)\) at all in Theorem 8.3. It strikes me as more logical to first prove the statement involving the internal direct product, to derive from this the statement about the external direct product, and only then to use the "\(U\)-groups" to illustrate the facts about direct product that Gallian presented in Chapter 8.

      There is a more common way that mathematicians show that the indicated map from \(U(st)\) to \(U(s)\oplus U(t)\) is onto. This surjectivity is a consequence of an important theorem not usually discussed in a first course on abstract algebra. This theorem has a name like no other in mathematics. I'm embarrassed to give it (and wonder if Gallian originally used it to prove Theorem 8.3 in an old edition of the book, but decided to remove it because of the name). In the English-speaking world, the theorem is called the Chinese Remainder Theorem. I kid you not; you can look it up on Wikipedia, where you can find both some history and a proof (look under "Existence (constructive proof)"). I have never heard this theorem called by any other name, but I rather doubt that this is what it's called in China. I apologize to anyone offended by the name.

    • The last few lines of p. 161 use facts proven by Gauss for which (I think) no proof appears in the book. This is consistent with the opening sentence of the section, "The \(U\)-groups provide a convenient way to illustrate the preceding ideas." (Emphasis added by me.) If the purpose of the section is simply to illustrate some usefulness of these ideas, without dotting every i and crossing every t, that's fine, but the author should make it clear what the student should be able to prove, and what the student should just take on faith for the sake of this illustration.
    Something that I'm surprised Gallian didn't give as a corollary to Theorem 8.3 is a multiplicativity property of Euler's phi-function: if \(s\) and \(t\) are relatively prime, then \(\phi(st)=\phi(s)\phi(t)\), with a similar statement for products of more than two pairwise-relatively-prime numbers. Since it's easy to see that \(\phi(p^n)=p^n-p^{n-1}\) for any prime power \(p^n\), this gives us an easy way to compute \(\phi(m)\) from the prime factorization of any given positive integer \(m\).

  • Ch. 8/ 1, 4, 5, 14, 15, 37, 48 (see note below), 51, 52, 53, 56, 59

      Note on #48. Interpret this as the following two-part problem. (a) For each automorphism \(\phi\in {\rm Aut}(\bfz_2\oplus \bfz_2)\), write down a formula for \(\phi\). (Thus, for each \(\phi\)—of which there are not very many—you should be writing down a formula of the form \(\phi(j,k)=(\mbox{[expression involving $j$ and/or $k$]}, \mbox{[another expression involving $j$ and/or $k$]})\).
      There are not a lot of these. These automorphisms, say \(\phi_1, \phi_2, \dots, \phi_N\), are the elements of \({\rm Aut}(\bfz_2\oplus \bfz_2)\); we have not yet incorporated any information about the group operation of \({\rm Aut}(\bfz_2\oplus \bfz_2)\). (b) Find a familiar \(N\)-element group to which \({\rm Aut}(\bfz_2\oplus \bfz_2)\) is isomorphic. For this, you may find the following ideas helpful:

        (i) The Klein group \(K\) is the four-element group \(\{e,a,b,c\}\), where \(a^2=b^2=c^2=e\) and the product of any two distinct order-two elements, in either order, is the third). Since an automorphism carries order-two elements to order-two elements, any automorphism of \(K\) must permute \(a,b, \mbox{and} c\). This fact should help you identify a familiar group \(G\) to which \({\rm Aut}(K)\) is isomorphic.

        (ii) As we have shown before, every four-element group is either cyclic (hence isomorphic to \(\bfz_4\)) or is isomorphic to \(K\); the group \(\bfz_2\oplus \bfz_2\) is of the latter type. Knowing that \({\rm Aut}(K)\approx G\), we can then use problem NB 5.2(b) to see that \({\rm Aut}(\bfz_2\oplus \bfz_2)\) is isomorphic to the same familiar group \(G\).

    No homework will be collected for Assignment 5.

    • Second midterm date The date for the second midterm will be Fri. Mar. 29.
    F 4/12/24 Assignment 6

  • Read Chapter 9.

  • Ch. 9/ 6, 11, 12, 13, 14, 17, 18, 34, 37, 43, 47, 51

    • Note for #6: Recall that Example 9 of Chapter 1 gave the formula for the inverse of an invertible \(2\times 2\) real matrix.

  • Read the handout One-to-one and onto: What you are really doing when you solve equations that I've added to the Miscellaneous Handouts page. This was written for a lower-level class, so the only equations in the handout are equations in a single real variable \(x\). But the same principles apply whether the variable are numbers, vectors, matrices, group elements, or elements of any other type of set.

  • Read my exam-solutions posted on Canvas. Compare my solutions to your own, observing which words, and word-order, I use in which situations. Make sure you understand everything in the blue comment "IMPORTANT" after the problem-6 solution. The principles go far beyond the example in the solution. Make sure you understand that these are the same principles as in the "What you are really doing when you solve equations" handout.

  • Do this non-book problem.

    On Friday 4/12 hand in only the following problems:

    • Ch. 9/ 6, 12, 14, 18, 34
    • Non-book problem NB 6.1cd.
    • W 4/24 Assignment 7

  • Read Chapter 10, except for Examples 16 and 17. (Those two examples are optional reading. I'm not holding you responsible for them.) Keep up with the following schedule, or get ahead of it:
    • Read up through Theorem 10.2 before the Friday 4/12 class.
      When you read Theorems 10.1 and 10.2 for homomorphisms, compare them with the analogous theorems in Chapter 6 for isomorphisms (Theorems 6.2 and 6.3).

    • Read up through Theorem 10.3 before the Monday 4/15 class.

    • Finish Chapter 10 (minus Examples 16 and 17) before the Wed. 4/17 class. (I am not holding you responsible for Examples 16 and 17.)

        Note: In the Corollary to the First Isomorphism Theorem on p. 201, the wording should have been "... from a finite group \(G\) to a finite group \(\overline{G}\)." We didn't cover this corollary in class, but you should know it and be able to prove it. The proof is a straightforward combination of earlier results; see the sentence above the corollary. The relevance of Lagrange's Theorem is something that I'm not sure was ever stated directly in the book or in class: when \(G\) is a finite group and \(H\) is a subgroup, not only does \(|H|\) divide \(|G|\), the index \(|G|/|H|\) also divides \(|G|\), since whenever an integer \(m\) is a divisor of an integer \(n\), the quotient \(m/n\) is another divisor of \(m\).

  • Ch. 10/ 2, 3, 4, 7, 8, 9, 11, 14, 16, 20, 23, 24, 27, 35, 48, 51, 64.
    See notes below on some of these exercises.

  • Do these non-book problems.

    Notes on some of the Chapter 10 exercises:

    • In #8, the last sentence should say (at least), "Why does this prove the result of   Exercise 23 of Chapter 5?" An exercise is not something that can be proven. Furthermore, better than "Why does this prove" would be "Why does this imply," or "How can this be used to prove;" the answer is an argument with more than one step.

    • In #14, the map should be written as \(x\mapsto 3x\ {\rm mod}\, 10\).

    • Regarding #20: (i) the difference between the first part of the question and the second is that "onto" does not appear in the second part; and (ii) non-book problem NB 7.1 is related to this exercise, but the two problems can be done independently and in either order.

    • In #48: To say that a group \(G'\) is a homomorphic image of a group \(G\) is to say that a homomorphism from \(G\) onto \(G'\). (In this particular problem, "onto" turns out not to make much difference, but I still want you to understand what the hypotheses were saying.)

  • Read Chapter 12 before the Monday 4/22 class.

    On Wednesday 4/24 hand in only the following problems:

    • Ch. 10/ 14, 16, 20, 24
    • Non-book problem NB 7.2
    • Any time before the final exam Assignment 8

  • Do these non-book problems. [Typo in NB 8.2 corrected 4/26/24.] I'm listing these first because problem NB 8.1 will make Chapter 12 exercises 1 and 3 a whole lot easier; it gives you a good hunting ground for examples.

  • Ch. 12/ 1, 2, 3, 4, 5, 6
    • None
  • In Canvas, under Files, I've posted a solution for non-book homework problem 7.2. In addition to a proof, I included some comments on notation (related to cosets and quotient/factor groups) that students often misunderstand or misuse when the subject is still new to them.

  • I was asked to post some extra review exercises, especially on earlier material. Listed below are some exercises from the book that I could have assigned for homework (in addition to the ones I assigned), if there were no limit to the amount of homework you could do, and/or if I hadn't already done them in class.

    Make sure you understand what that last sentence says and does not say. I am not promising that every type of problem you could see on the final exam is represented in the list below. Nor am I promising any relation between how many exercises from a given chapter are on the list, and that chapter's representation on the final exam.

    • Chapters 1–4: I won't have time to post any extra exercises from these chapters. However, even if I did have time to look through all those exercises, I would probably add relatively few of them to the list below. You now have more powerful tools to approach the earlier chapters' exercises that you had when homework from those chapters was assigned.

    • Ch. 5/ 8, 16, 32, 55, 58

    • Ch. 6/ 1, 12, 18, 33, 34, 38

    • Ch. 7/ 4, 5, 7, 10, 21, 22, 36, 38, 42, 43

    • Ch. 8/ 3, 7, 8, 16, 17, 18, 19, 36. In 7, 16, and 17, the "general case" means the direct product of an arbitrary (finite) number of groups. "State the general case" means more than conjecture the general case; it means that you could write down a proof if you had to.

    • Ch. 9/ 1, 2, 10a, 39, 40, 55, 56, 61 (consider the element \(xH\in G/H\)), 70

    • Ch. 10/ 5, 12, 13, 15, 21, 38, 41 (the theorem that I didn't remember had this name!), 56, 59 (don't overlook the word "onto" in the hypothesis!)

    • Ch. 12/ 10, 46, 13 ("describe" here means "figure them all out, explicitly"), 40, 41 (the answers for 40 and 41 are different!), 49. I listed #46 out of order because #46 becomes almost pointless once you've correctly answered #13.
    • Thurs 5/2/24

    Final Exam
          Location: Our usual classroom
          Starting time: 7:30 a.m.

    Yes, that's 7:30 in the &*!@#$! morning :(

    I would hate to have to take an exam at that ungodly hour, and you have my sympathies (unless you're an early bird, bright and chipper and the envy of your classmates at that hour). As we discussed in class, circumstances beyond any of our control make it not possible to move the exam to a later hour. I'll probably set four alarms instead of my usual two to make sure I get to the exam on time; you do whatever you have to do.

    "Homework": Figure out what time you'll need to go to bed to get a good night's sleep, and do your best to keep to that schedule.

      For the glass-half-full folks: in my experience, a couple of extra hours of sleep is always worth more than a couple of extra hours of studying.

      For the glass-half-empty folks: if it's half a day before the exam, and you still don't know what you need to know, it's too late to fix that. Cut your losses and give yourself the chance to show what you do know.

    That's not the oracle speaking; it's just Dad's best advice. We're not all built the same way, and you know yourselves better than I do. But at least do your best to start your review enough days in advance that you won't feel pressed to study when you should be sleeping.

    And the night before the exam, if you find yourself wanting to bind and gag that noisy roommate and throw him/her in the closet, or wanting to take an axe to all the sound-systems and TVs in adjoining apartments, you didn't get those ideas from me. Nope, I never wanted to do any of that.

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