Tour:Lecture transcript one (beginners)

From Groupprops
Jump to: navigation, search
This page is a Lecture transcript page, part of the Groupprops guided tour for beginners (Jump to beginning of tour)
UP: Introduction one |
NEXT SECTION Lecture transcript: Lecture transcript two
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part

This page gives a lecture transcript for part one of the guided tour. The lecture transcript can be read along with the guided tour, and can be used by lecturers to create lectures using the guided tour.

This lecture transcript may not be best for reviewing part one for people who already understand it thoroughly; however, it may be helpful for people who prefer a flow format to a modular presentation.

We are now on the following page of the tour: Tour:group

You've probably heard the word group in ordinary English. Group means some kind of collection of people or things chosen to work together or act together. Now, the ordinary English definition of group is pretty fuzzy. I mean, there's no real criterion to definitively decide whether something is group. So is a bunch of toffees a group? Is a collection of students in a class a group? There's no clear line.

We mathematicians also use the term group for a kind of collection, but we have a very precise notion of what a group is. The mathematical definition of group means the same thing to different people. So if a mathematician is presented with something, he or she can definitely answer whether that's a group or not.

The mathematician's group is a set (a precise mathematical term for a reasonably well-defined collection of things). But it's not just a set. This set is endowed with a binary operation, or a multiplication or product. Basically, a rule that takes in two elements of the set (in order) and gives out an element of the set. Let's make this precise with notation.

Chalkboard (1,1):
A group is a set G with a binary operation * (termed the multiplication or product ...

But we don't just want any set with a binary operation to be declared a group. Rather, we want some stringent conditions on the binary operation, that give some measure of well-behavedness.

Here's the complete definition:

Chalkboard (1,1):

A group is a set G with a binary operation * (termed the multiplication or product) such that the following hold:

  • For any a,b,c in G, a * (b * c) = (a * b) * c. This property is termed associativity.
  • There exists an element e in G such that a * e = e * a = a for all a in G. Such an e is termed a neutral element or identity element for G.
  • For any a in G, there is an element b such that a * b = b * a = e. Such a b is termed an inverse of a and is denoted as a^{-1}.

So a group is a set with a binary operation satisfying a condition called associativity, an identity element, and inverses.

Now, what we've essentially done here is take a set. Any set. Then we gave it a binary operation. If that binary operation satisfies certain conditions, it is a group.

Some people prefer to define groups in a slightly different way. They want to put the identity element and the inverses as part of the structure. So, rather than specifying just the binary operation, they want to spell out explicitly what the identity element is, and how inverses should be computed. So this new definition of group looks like:

Chalkboard (1,2):

A group is a set G equipped with three operations:

  • A binary operation * (infix operator) termed the multiplication or product
  • A unary operation {}^{-1} (superscript operator) termed the inverse map
  • A 0-ary operation which gives a constant element, denoted by e, termed the identity element or neutral element

satisfying the following three compatibility conditions:

What's the difference between these two definitions? Essentially, a difference in what you specify beforehand. In one definition, only the binary operation is given, and you're on your own in trying to figure out what the identity element and inverses are. In the other, you are given the identity element and inverses. It's like defining a kingdom without specifying its king, as opposed to with specifying its king. We'll see later why these definitions are equivalent. Essentially, it boils down to saying that the binary operation leaves you with no choice about the identity element and inverses: they're completely determined by it.

One more thing to note: in mathematics, it is common to give multiple definitions that are equivalent, that all mean the same thing. This is in sharp contrast to some other subjects, where the same definition could mean different things.

Okay, by now some of you might be asking: what are some examples of groups? After all, what I've done is bad pedagogy. I gave a definition without giving examples. Ideally, I should have given you five examples, then observed common features between them, and naturally come up with the definition of group.

The really interesting thing about mathematics, though, is that it flows both ways: from the definition (or abstraction) to the examples (or concrete stuff) and from the examples to the definition. People come up with definitions and then ask: are there any examples of this? Or they may look at some common concrete situations and say: what new concept can I make out of this? And in mathematics, we can give a definition without any examples and still make sure that any two mathematicians will understand the same thing from it, because the language of mathematics is fairly precise. So the definition of group is all there is to know about group. (Philosophers call such definitions intensional definitions -- definitions that are complete in themselves)

But there's another reason why I didn't give you examples -- all the three rules: associativity, identity element (or neutral element) and inverses, are probably already familiar to you. You've seen them for numbers -- rational numbers, real numbers. So you're in a position to come up with some examples yourself.

So what we need to do is look at the laws of associativity, identity element and inverses that hold for number systems like rationals, reals, and complex numbers, and see exactly what they say about things being groups.

Let's first zoom in to the real numbers under addition. The set G above is \R, and the binary operation * is +. Associativity? Sure, we learned that in kindergarten. Identity element? That's zero. Inverses? That's the additive inverse, the negative of a number.

So the upshot:

Chalkboard (2,1):

Examples of groups:

  • The set of real numbers forms a group under addition. This group is denoted (\R,+).

What about multiplication? Associativity? Yeah, sure. Identity element? One, what else? Inverses? Yeah, sure... but no, wait. 0 does not have a multiplicative inverse.

So the reals do not form a group under multiplication. Now, there's no fussing around here about it being very close and so may be we can call it a group. If it doesn't satisfy the conditions, it isn't a group.

Well, here there could be things you might think of trying. First, you might want to throw the troublesome thing out. So 0 gets thrown out, and you have a group:

Chalkboard (2,1):

Examples of groups:

  • The set of real numbers forms a group under addition. This group is denoted (\R,+).
  • The set of nonzero real numbers forms a group under multiplication.

The other thing you might think of trying is to add in an inverse for zero. Remember, these things don't have to be actual numbers. So you might want to add in another element called \infty, that is defined as 1/0. Unfortunately, if you try doing that, you mess up other stuff. That requires a bit of thinking, and we'll get to it later, but the upshot is that associativity gets compromised.

So the reals under addition form a group, and the nonzero reals under multiplication form a group. Similarly, you can see that the integers under addition form a group. The rational numbers under addition form a group. The nonzero rational numbers under multiplication form a group.

Because in each of the cases, we have associativity, we have an identity element (0 for addition, 1 for multiplication) and we have inverses (the additive inverse is the negative, the multiplicative inverse is, well, the multiplicative inverse).

Now what about some non-groups? So are there sets with binary operations that don't form groups? There could be three problems: no associativity, no identity, no inverses. So, for instance, the real numbers aren't a group under multiplication because 0 doesn't have an inverse. The positive integers aren't a group unde raddition because there's no identity element (zero isn't positive, so it's not in the set per se).
So groups are characterized by a binary operation with associativity, identity elements and inverses. If any of these fail, what you have isn't a group. May be it's close. But if it isn't a group, it isn't a group. You need all three things: associativity, identity elements, and inverses, to form a group.
We are now on the following page of the tour: Tour:abelian group

Okay, before going further, I'd like to point out some things to you about the examples I gave of groups. The first is that they're all infinite in size. The second is that in all the cases, the binary operation was commutative. That's because of the commutativity law for addition and multiplication of real numbers.

So this could lead to a question: does the definition of a group force the binary operation to be commutative? Or can we find groups where the binary operation isn't commutative? Either way, it would be cool. In the first case, we'd have something strange: commutativity can be derived just from associativity, identity element and inverses. In the second case, we'd have sets with weird operations that don't commute, but are still nice enough to associate, have identity elements, and have inverses.

It turns out that we can find groups where the binary operation is not commutative. In fact, a lot of the interesting groups are of that sort. So we want a name to distinguish those groups where the binary operation is commutative, because they're special and, in some sense, much easier to study. We call these abelian groups.

Chalkboard (1,1):

A group is a set G with a binary operation * (termed the multiplication or product) such that the following hold:

  • For any a,b,c in G, a * (b * c) = (a * b) * c. This property is termed associativity.
  • There exists an element e in G such that a * e = e * a = a for all a in G. Such an e is termed a neutral element or identity element for G.
  • For any a in G, there is an element b such that a * b = b * a = e. Such a b is termed an inverse of a and is denoted as a^{-1}.
An abelian group is a group where any two elements commute. In other words, G is an abelian group if ab = ba for all a,b \in G.

And we call groups that aren't abelian (guess what?) non-abelian. And now you probably want an example of a non-abelian group. And I'm not going to give you an example right away. I could, they're not hard. But my purpose right now is to make you appreciate that one could define something and work with it even without having an example in mind. May be there aren't any examples, or may be there are and we don't know how to construct them, but one could still prove theorems about it. And why can we do this in mathematics? It is the preciseness and certainty of mathematical language that allows us to explore totally strange lands.

We are now on the following page of the tour: Tour:subgroup

So now it's time to switch track a bit and look at the notion of subgroup.

You know what a subset of a set is. So what is a subgroup? Intuitively, it might be a subset that also happens to be a group. But there's something more. We want the group structure on the subset to be related to the group structure on the whole set. So we want the product to be the same.

This is a place where we branch off two possibilities, based on the two definitions of the group. (Points at chalkboard (1,1)). The first definition starts with a binary operation, the multiplication, and considers the inverse map and identity element to be incidental to the structure. So in that definition, a subgroup is a subset that happens to be closed under the binary operation, and becomes a group under that induced binary operation. On the other hand, in the second definition, we want the same identity element and same inverse map in the group and in the subgroup. We want it to inherit more of the structure rather than the possibility of it getting its own identity element and its own inverse map.

Let's put down these definitions:


Now, it turns out that both these definitions are equal. But that's not something I want you to take for granted or consider a triviality. It has a lot to do with the way we've defined groups and the structural interplay of associativity, identity element, and inverses. In fact, if you perturbed the definition of group just a little bit, you'd get two notions of substructures that would actually have different meanings.

Now, I'll write down a third definition of subgroup. This definition isn't extremely important in and of itself. I think it's useful to drive the point that the same concept can be defined in many different ways. We'll look at this definition again later.


We are now on the following page of the tour: Tour:trivial group

And now it's time to give you the most important example of a group.

So once a mathematics student was asked to give an example of a set, and she said the empty set. So the teacher was frustrated, and asked the mathematics student for another example of a set. So she said, the one-point set. The teacher asked for more examples, and the student said, "Well, there are lots of examples, but these are the ones that I can say off the top of my head."

Examples have lots of roles to play, and while one role is to give interesting and representative things, another is to give the extreme cases. The empty set and the one-point set are two extreme examples (for trivial as well as more sophisticated reasons). These examples set a boundary, and they relate to all the other examples in fundamental ways. In groups, there is one extreme example, and that's the trivial group.

A trivial group is a group with just one element.

We are now on the following page of the tour: Tour:Verifying the group axioms