Tag Archives: coset

Introducing abstract topics

I have been busy for the past several years revising abstractmath.org (abmath). Now I believe, perhaps foolishly, that most of the articles in abmath have reached beta, so now it is time for something new.

For some time I have been considering writing introductions to topics in abstract math, some typically studied by undergraduates and some taken by scientists and engineers. The topics I have in mind to do first include group theory and category theory.

The point of these introductions is to get the student started at the very beginning of the topic, when some students give up in total confusion. They meet and fall off of what I have called the abstraction cliff, which is discussed here and also in my blog posts Very early difficulties and Very early difficulties II.

I may have stolen the phrase “abstraction cliff” from someone else.

Group theory

Group theory sets several traps for beginning students.

Multiplication table

  • A student may balk when a small finite group is defined using a set of letters in a multiplication table.
    “But you didn’t say what the letters are or what the multiplication is?”
  • Such a definition is an abstract definition, in contrast to the definition of “prime”, for example, which is stated in terms of already known entities, namely the integers.
  • The multiplication table of a group tells you exactly what the binary operation is and any set with an operation that makes such a table correct is an example of the group being defined.
  • A student who has no understanding of abstraction is going to be totally lost in this situation. It is quite possible that the professor has never even mentioned the concept of abstract definition. The professor is probably like most successful mathematicians: when they were students, they understood abstraction without having to have it explained, and possibly without even noticing they did so.

Cosets

  • Cosets are a real killer. Some students at this stage are nowhere near thinking of a set as an object or a thing. The concept of applying a binary operation on a pair of sets (or any other mathematical objects with internal structure) is completely foreign to them. Did anyone ever talk to them about mathematical objects?
  • The consequence of this early difficulty is that such a student will find it hard to understand what a quotient group is, and that is one of the major concepts you get early in a group theory course.
  • The conceptual problems with multiplication of cosets is similar to those with pointwise addition of functions. Given two functions $f,g:\mathbb{R}\to\mathbb{R}$, you define $f+g$ to be the function \[(f+g)(x):=f(x)+g(x)\] Along with pointwise multiplication, this makes the space of functions $\mathbb{R}\to\mathbb{R}$ a ring with nice properties.
  • But you have to understand that each element of the ring is a function thought of as a single math object. The values of the function are properties of the function, but they are not elements of the ring. (You can include the real numbers in the ring as constant functions, but don’t confuse me with facts.)
  • Similarly the elements of the quotient group are math objects called cosets. They are not elements of the original group. (To add to the confusion, they are also blocks of a congruence.)

Isomorphic groups

  • Many books, and many professors (including me) regard two isomorphic groups as the same. I remember getting anguished questions: “But the elements of $\mathbb{Z}_2$ are equivalence classes and the elements of the group of permutations of $\{1,2\}$ are functions.”
  • I admit that regarding two isomorphic groups as the same needs to be treated carefully when, unlike $\mathbb{Z}_2$, the group has a nontrivial automorphism group. ($\mathbb{Z}_3$ is “the same as itself” in two different ways.) But you don’t have to bring that up the first time you attack that subject, any more than you have to bring up the fact that the category of sets does not have a set of objects on the first day you define categories.

Category theory

Category theory causes similar troubles. Beginning college math majors don’t usually meet it early. But category theory has begun to be used in other fields, so plenty of computer science students, people dealing with databases, and so on are suddenly trying to understand categories and failing to do so at the very start.

The G&G post A new kind of introduction to category theory constitutes an alpha draft of the first part of an article introducing category theory following the ideas of this post.

Objects and arrows are abstract

  • Every once in a while someone asks a question on Math StackExchange that shows they have no idea that an object of a category need not have elements and that morphisms need not be functions that take elements to elements.
  • One questioner understood that the claim that a morphism need not be a function meant that it might be a multivalued function.

Duality

  • That misunderstanding comes up with duality. The definition of dual category requires turning the arrows around. Even if the original morphism takes elements to elements, the opposite morphism does not have to take elements to elements. In the case of the category of sets, an arrow in $\text{Set}^{op}$ cannot take elements to elements — for example, the opposite of the function $\emptyset\to\{1,2\}$.
  • The fact that there is a concrete category equivalent to $\text{Set}^{op}$ is a red herring. It involves different sets: the function corresponding to the function just mentioned goes from a four-element set to a singleton. But in the category $\text{Set}^{op}$ as defined it is simply an arrow, not a function.

Not understanding how to use definitions

  • Some of the questioners on Math Stack Exchange ask how to prove a statement that is quite simple to prove directly from the definitions of the terms involved, but what they ask and what they are obviously trying to do is to gain an intuition in order to understand why the statement is true. This is backward — the first thing you should do is use the definition (at least in the first few days of a math class — after that you have to use theorems as well!
  • I have discussed this in the blog post Insights into mathematical definitions (which gives references to other longer discussions by math ed people). See also the abmath section Rewrite according to the definitions.

How an introduction to a math topic needs to be written

The following list shows some of the tactics I am thinking of using in the math topic introductions. It is quite likely that I will conclude that some tactics won’t work, and I am sure that tactics I haven’t mentioned here will be used.

  • The introductions should not go very far into the subject. Instead, they should bring an exhaustive and explicit discussion of how to get into the very earliest part of the topic, perhaps the definition, some examples, and a few simple theorems. I doubt that a group theory student who hasn’t mastered abstraction and what proofs are about will ever be ready to learn the Sylow theorems.
  • You can’t do examples and definitions simultaneously, but you can come close by going through an example step by step, checking each part of the definition.
  • There is a real split between students who want the definitions first
    (most of whom don’t have the abstraction problems I am trying to overcome)
    and those who really really think they need examples first (the majority)
    because they don’t understand abstraction.

  • When you introduce an axiom, give an example of how you would prove that some binary operation satisfies the axiom. For example, if the axiom is that every element of a group must have an inverse, right then and there prove that addition on the integers satisfies the axiom and disprove that multiplication on integers satisies it.
  • When the definition uses some undefined math objects, point out immediately with examples that you can’t have any intuition about them except what the axioms give you. (In contrast to definition of division of integers, where you and the student already have intuitions about the objects.)
  • Make explicit the possible problems with abstractmath.org and Gyre&Gimble) will indeed find it difficult to become mathematical researchers — but not impossible!
  • But that is not the point. All college math professors will get people who will go into theoretical computing science, and therefore need to understand category theory, or into particle physics, and need to understand groups, and so on.
  • By being clear at the earliest stages of how mathematicians actually do math, they will produce more people in other fields who actually have some grasp of what is going on with the topics they have studied in math classes, and hopefully will be willing to go back and learn some more math if some type of math rears its head in the theories of their field.
  • Besides, why do you want to alienate huge numbers of people from math, as our way of teaching in the past has done?
  • “Our” means grammar school teachers, high school teachers and college professors.

Acknowledgment

Thanks to Kevin Clift for corrections.

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A very early satori that occurs with beginning abstract math students

In the previous post Pattern recognition and me, I wrote about how much I enjoyed sudden flashes of understanding that were caused by my recognizing a pattern (or learning about a pattern). I have had several such, shall we say, Thrills in learning about math and doing research in math. This post is about a very early thrill I had when I first started studying abstract algebra. As is my wont, I will make various pronouncements about what these mean for teaching and understanding math.

Cosets

Early in any undergraduate course involving group theory, you learn about cosets.

Basic facts about cosets

  1. Every subgroup of a group generates a set of left cosets and a set of right cosets.
  2. If $H$ is a subgroup of $G$ and $a$ and $b$ are elements of $G$, then $a$ and $b$ are in the same left coset of $H$ if and only if $a^{-1}b\in H$. They are in the same right coset of $H$ if and only if $ab^{-1}\in H$.
  3. Alternative definition: $a$ and $b$ are in the same left coset of $H$ if $a=bh$ for some $h\in H$ and are in the same right coset of $H$ if $a=hb$ for some $h\in H$
  4. One of the (left or right) cosets of $H$ is $H$ itself.
  5. The relations
    $a\underset{L}\sim b$ if and only if $a^{-1}b\in H$

    and

    $a\underset{R}\sim b$ if and only if $ab^{-1}\in H$

    are equivalence relations.

  6. It follows from (5) that each of the set of left cosets of $H$ and the set of right cosets of $H$ is a partition of $G$.
  7. By definition, $H$ is a normal subgroup of $G$ if the two sets of cosets coincide.
  8. The index of a subgroup in a group is the cardinal number of (left or right) cosets the subgroup has.

Elementary proofs in group theory

In the course, you will be asked to prove some of the interrelationships between (2) through (5) using just the definitions of group and subgroup. The teacher assigns these exercises to train the students in the elementary algebra of elements of groups.

Examples:

  1. If $a=bh$ for some $h\in H$, then $b=ah’$ for some $h’\in H$. Proof: If $a=bh$, then $ah^{-1}=(bh)h^{-1}=b(hh^{-1})=b$.
  2. If $a^{-1}b\in H$, then $b=ah$ for some $h\in H$. Proof: $b=a(a^{-1}b)$.
  3. The relation “$\underset{L}\sim$” is transitive. Proof: Let $a^{-1}b\in H$ and $b^{-1}c\in H$. Then $a^{-1}c=a^{-1}bb^{-1}c$ is the product of two elements of $H$ and so is in $H$.
Miscellaneous remarks about the examples
  • Which exercises are used depends on what is taken as definition of coset.
  • In proving Exercise 2 at the board, the instructor might write “Proof: $b=a(a^{-1}b)$” on the board and the point to the expression “$a^{-1}b$” and say, “$a^{-1}b$ is in $H$!”
  • I wrote “$a^{-1}c=a^{-1}bb^{-1}c$” in Exercise 3. That will result in some brave student asking, “How on earth did you think of inserting $bb^{-1}$ like that?” The only reasonable answer is: “This is a trick that often helps in dealing with group elements, so keep it in mind.” See Rabbits.
  • That expression “$a^{-1}c=a^{-1}bb^{-1}c$” doesn’t explicitly mention that it uses associativity. That, too, might cause pointing at the board.
  • Pointing at the board is one thing you can do in a video presentation that you can’t do in a text. But in watching a video, it is harder to flip back to look at something done earlier. Flipping is easier to do if the video is short.
  • The first sentence of the proof of Exercise 3 is, “Let $a^{-1}b\in H$ and $b^{-1}c\in H$.” This uses rewrite according to the definition. One hopes that beginning group theory students already know about rewrite according to the definition. But my experience is that there will be some who don’t automatically do it.
  • in beginning abstract math courses, very few teachers
    tell students about rewrite according to the definition. Why not?

  • An excellent exercise for the students that would require more than short algebraic calculations would be:
    • Discuss which of the two definitions of left coset embedded in (2), (3), (5) and (6) is preferable.
    • Show in detail how it is equivalent to the other definition.

A theorem

In the undergraduate course, you will almost certainly be asked to prove this theorem:

A subgroup $H$ of index $2$ of a group $G$ is normal in $G$.

Proving the theorem

In trying to prove this, a student may fiddle around with the definition of left and right coset for awhile using elementary manipulations of group elements as illustrated above. Then a lightbulb appears:

In the 1980’s or earlier a well known computer scientist wrote to me that something I had written gave him a satori. I was flattered, but I had to look up “satori”.

If the subgroup has index $2$ then there are two left cosets and two right cosets. One of the left cosets and one of the right cosets must be $H$ itself. In that case the left coset must be the complement of $H$ and so must the right coset. So those two cosets must be the same set! So the $H$ is normal in $G$.

This is one of the earlier cases of sudden pattern recognition that occurs among students of abstract math. Its main attraction for me is that suddenly after a bunch of algebraic calculations (enough to determine that the cosets form a partition) you get the fact that the left cosets are the same as the right cosets by a purely conceptual observation with no computation at all.

This proof raises a question:

Why isn’t this point immediately obvious to students?

I have to admit that it was not immediately obvious to me. However, before I thought about it much someone told me how to do it. So I was denied the Thrill of figuring this out myself. Nevertheless I thought the solution was, shall we say, cute, and so had a little thrill.

A story about how the light bulb appears

In doing exercises like those above, the student has become accustomed to using algebraic manipulation to prove things about groups. They naturally start doing such calculations to prove this theorem. They presevere for awhile…

Scenario I

Some students may be in the habit of abandoning their calculations, getting up to walk around, and trying to find other points of view.

  1. They think: What else do I know besides the definitions of cosets?
  2. Well, the cosets form a partition of the group.
  3. So they draw a picture of two boxes for the left cosets and two boxes for the right cosets, marking one box in each as being the subgroup $H$.
  4. If they have a sufficiently clear picture in their head of how a partition behaves, it dawns on them that the other two boxes have to be the same.
Remarks about Scenario I
  • Not many students at the earliest level of abstract math ever take a break and walk around with the intent of having another approach come to mind. Those who do Will Go Far. Teachers should encourage this practice. I need to push this in abstractmath.org.
  • In good weather, David Hilbert would stand outside at a shelf doing math or writing it up. Every once in awhile he would stop for awhile and work in his garden. The breaks no doubt helped. So did standing up, I bet. (I don’t remember where I read this.)
  • This scenario would take place only if the students have a clear understanding of what a partition is. I suspect that often the first place they see the connection between equivalence relations and partitions is in a hasty introduction at the beginning of a group theory or abstract algebra course, so the understanding has not had long to sink in.

Scenario II

Some students continue to calculate…

  1. They might say, suppose $a$ is not in $H$. Then it is in the other left coset, namely $aH$.
  2. Now suppose $a$ is not in the “other” right coset, the one that is not $H$. But there are only two right cosets, so $a$ must be in $H$.
  3. But that contradicts the first calculation I made, so the only possibility left is that $a$ is in the right coset $Ha$. So $aH\subseteq Ha$.
  4. Aha! But then I can use the same argument the other way around, getting $Ha\subseteq aH$.
  5. So it must be that $aH=Ha$. Aha! …indeed.
Remarks about Scenario 2
  • In step (2), the student is starting a proof by contradiction. Many beginning abstract math students are not savvy enough to do this.
  • Step (4) involves recognizing that an argument has a dual. Abstractmath.org does not mention dual arguments and I can’t remember emphasizing the idea to my classes. Tsk.
  • Scenario 2 involves the student continuing algebraic calculations till the lightbulb strikes. The lightbulb could also occur in other places in the calculation.

References

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