The Leap Day Mathematics A to Z term today is another abstract algebra term. This one again comes from from Gaurish, chief author of the Gaurish4Math blog. Part of it is going to be easy. Part of it is going to need a running start.
The “subgroup” part of this is easy. Remember that a “group” means a collection of things and some operation that lets us combine them. We usually call that either addition or multiplication. We usually write it out like it’s multiplication. If a and b are things from the collection, we write “ab” to mean adding or multiplying them together. (If we had a ring, we’d have something like addition and something like multiplication, and we’d be able to do “a + b” or “ab” as needed.)
So with that in mind, the first thing you’d imagine a subgroup to be? That’s what it is. It’s a collection of things, all of which are in the original group, and that uses the same operation as the original group. For example, if the original group has a set that’s the whole numbers and the operation of addition, a subgroup would be the even numbers and the same old addition.
Now things will get much clearer if I have names. Let me use G to mean some group. This is a common generic name for a group. Let me use H as the name for a subgroup of G. This is a common generic name for a subgroup of G. You see how deeply we reach to find names for things. And we’ll still want names for elements inside groups. Those are almost always lowercase letters: a and b, for example. If we want to make clear it’s something from G’s set, we might use g. If we want to be make clear it’s something from H’s set, we might use h.
I need to tax your imagination again. Suppose “g” is some element in G’s set. What would you imagine the symbol “gH” means? No, imagine something simpler.
Mathematicians call this “left-multiplying H by g”. What we mean is, take every single element h that’s in the set H, and find out what gh is. Then take all these products together. That’s the set “gH”. This might be a subgroup. It might not. No telling. Not without knowing what G is, what H is, what g is, and what the operation is. And we call it left-multiplying even if the operation is called addition or something else. It’s just easier to have a standard name even if the name doesn’t make perfect sense.
That we named something left-multiplying probably inspires a question. Is there right-multiplying? Yes, there is. We’d write that as “Hg”. And that means take every single element h that’s in the set H, and find out what hg is. Then take all these products together.
You see the subtle difference between left-multiplying and right-multiplying. In the one, you multiply everything in H on the left. In the other, you multiply everything in H on the right.
So. Take anything in G. Let me call that g. If it’s always, necessarily, true that the left-product, gH, is the same set as the right-product, Hg, then H is a normal subgroup of G.
The mistake mathematics majors make in doing this: we need the set gH to be the same as the set Hg. That is, the whole collection of products has to be the same for left-multiplying as right-multiplying. Nobody cares whether for any particular thing, h, inside H whether gh is the same as hg. It doesn’t matter. It’s whether the whole collection of things is the same that counts. I assume every mathematics major makes this mistake. I did, anyway.
The natural thing to wonder here: how can the set gH ever not be the same as Hg? For that matter, how can a single product gh ever not be the same as hg? Do mathematicians just forget how multiplication works?
Technically speaking no, we don’t. We just want to be able to talk about operations where maybe the order does too matter. With ordinary regular-old-number addition and multiplication the order doesn’t matter. gh always equals hg. We say this “commutes”. And if the operation for a group commutes, then every subgroup is a normal subgroup.
But sometimes we’re interested in things that don’t commute. Or that we can’t assume commute. The example every algebra book uses for this is three-dimensional rotations. Set your algebra book down on a table. If you don’t have an algebra book you may use another one instead. I recommend Christopher Miller’s American Cornball: A Laffopedic Guide To The Formerly Funny. It’s a fine guide to all sorts of jokes that used to amuse and what was supposed to be amusing about them. If you don’t have a table then I don’t know what to suggest.
Spin the book clockwise on the table and then stand it up on the edge nearer you. Then try again. Put the book back where it started. Stand it up on the edge nearer you and then spin it clockwise on the table. The book faces a different way this time around. (If it doesn’t, you spun too much. Try again until you get the answer I said.)
Three-dimensional rotations like this form a group. The different ways you can turn something are the elements of its set. The operation between two rotations is just to do one and then the other, in order. But they don’t commute, not most of the time. So they can have a subgroup that isn’t normal.
You may believe me now that such things exist. Now you can move on to wondering why we should care.
Let me start by saying every group has at least two normal subgroups. Whatever your group G is, there’s a subgroup that’s made up just of the identity element and the group’s operation. The identity element is the thing that acts like 1 does for multiplication. You can multiply stuff by it and you get the same thing you started. The identity and the operator make a subgroup. And you’ll convince yourself that it’s a normal subgroup as soon as you write down g1 = 1g.
(Wait, you might ask! What if multiplying on the left has a different identity than multiplying on the right does? Great question. Very good insight. You’ve got a knack for asking good questions. If we have that then we’re working with a more exotic group-like mathematical object, so don’t worry.)
So the identity, ‘1’, makes a normal subgroup. Here’s another normal subgroup. The whole of G qualifies. (It’s OK if you feel uneasy. Think it over.)
So ‘1’ is a normal subgroup of G. G is a normal subgroup of G. They’re boring answers. We know them before we even know anything about G. But they qualify.
Does this sound familiar any? We have a thing. ‘1’ and the original thing subdivide it. It might be possible to subdivide it more, but maybe not.
Is this all … factoring?
Please here pretend I make a bunch of awkward faces while trying not to say either yes or no. But if H is a normal subgroup of G, then we can write something G/H, just like we might write 4/2, and that means something.
That G/H we call a quotient group. It’s a subgroup, sure. As to what it is … well, let me go back to examples.
Let’s say that G is the set of whole numbers and the operation of ordinary old addition. And H is the set of whole numbers that are multiples of 4, again with addition. So the things in H are 0, 4, 8, 12, and so on. Also -4, -8, -12, and so on.
Suppose we pick things in G. And we use the group operation on the set of things in H. How many different sets can we get out of it? So for example we might pick the number 1 out of G. The set 1 + H is … well, list all the things that are in H, and add 1 to them. So that’s 1 + 0, 1 + 4, 1 + 8, 1 + 12, and 1 + -4, 1 + -8, 1 + -12, and so on. All told, it’s a bunch of numbers one more than a whole multiple of 4.
Or we might pick the number 7 out of G. The set 7 + H is 7 + 0, 7 + 4, 7 + 8, 7 + 12, and so on. It’s also got 7 + -4, 7 + -8, 7 + -12, and all that. These are all the numbers that are three more than a whole multiple of 4.
We might pick the number 8 out of G. This happens to be in H, but so what? The set 8 + H is going to be 8 + 0, 8 + 4, 8 + 8 … you know, these are all going to be multiples of 4 again. So 8 + H is just H. Some of these are simple.
How about the number 3? 3 + H is 3 + 0, 3 + 4, 3 + 8, and so on. The thing is, the collection of numbers you get by 3 + H is the same as the collection of numbers you get by 7 + H. Both 3 and 7 do the same thing when we add them to H.
Fiddle around with this and you realize there’s only four possible different sets you get out of this. You can get 0 + H, 1 + H, 2 + H, or 3 + H. Any other numbers in G give you a set that looks exactly like one of those. So we can speak of 0, 1, 2, and 3 as being a new group, the “quotient group” that you get by G/H. (This looks more like remainders to me, too, but that’s the terminology we have.)
But we can do something like this with any group and any normal subgroup of that group. The normal subgroup gives us a way of picking out a representative set of the original group. That set shows off all the different ways we can manipulate the normal subgroup. It tells us things about the way the original group is put together.
Normal subgroups are not just “factors, but for groups”. They do give us a way to see groups as things built up of other groups. We can see structures in sets of things.