A Leap Day 2016 Mathematics A To Z: Isomorphism


Gillian B made the request that’s today’s A To Z word. I’d said it would be challenging. Many have been, so far. But I set up some of the work with “homomorphism” last time. As with “homomorphism” it’s a word that appears in several fields and about different kinds of mathematical structure. As with homomorphism, I’ll try describing what it is for groups. They seem least challenging to the imagination.

Isomorphism.

An isomorphism is a kind of homomorphism. And a homomorphism is a kind of thing we do with groups. A group is a mathematical construct made up of two things. One is a set of things. The other is an operation, like addition, where we take two of the things and get one of the things in the set. I think that’s as far as we need to go in this chain of defining things.

A homomorphism is a mapping, or if you like the word better, a function. The homomorphism matches everything in a group to the things in a group. It might be the same group; it might be a different group. What makes it a homomorphism is that it preserves addition.

I gave an example last time, with groups I called G and H. G had as its set the whole numbers 0 through 3 and as operation addition modulo 4. H had as its set the whole numbers 0 through 7 and as operation addition modulo 8. And I defined a homomorphism φ which took a number in G and matched it the number in H which was twice that. Then for any a and b which were in G’s set, φ(a + b) was equal to φ(a) + φ(b).

We can have all kinds of homomorphisms. For example, imagine my new φ1. It takes whatever you start with in G and maps it to the 0 inside H. φ1(1) = 0, φ1(2) = 0, φ1(3) = 0, φ1(0) = 0. It’s a legitimate homomorphism. Seems like it’s wasting a lot of what’s in H, though.

An isomorphism doesn’t waste anything that’s in H. It’s a homomorphism in which everything in G’s set matches to exactly one thing in H’s, and vice-versa. That is, it’s both a homomorphism and a bijection, to use one of the terms from the Summer 2015 A To Z. The key to remembering this is the “iso” prefix. It comes from the Greek “isos”, meaning “equal”. You can often understand an isomorphism from group G to group H showing how they’re the same thing. They might be represented differently, but they’re equivalent in the lights you use.

I can’t make an isomorphism between the G and the H I started with. Their sets are different sizes. There’s no matching everything in H’s set to everything in G’s set without some duplication. But we can make other examples.

For instance, let me start with a new group G. It’s got as its set the positive real numbers. And it has as its operation ordinary multiplication, the kind you always do. And I want a new group H. It’s got as its set all the real numbers, positive and negative. It has as its operation ordinary addition, the kind you always do.

For an isomorphism φ, take the number x that’s in G’s set. Match it to the number that’s the logarithm of x, found in H’s set. This is a one-to-one pairing: if the logarithm of x equals the logarithm of y, then x has to equal y. And it covers everything: all the positive real numbers have a logarithm, somewhere in the positive or negative real numbers.

And this is a homomorphism. Take any x and y that are in G’s set. Their “addition”, the group operation, is to multiply them together. So “x + y”, in G, gives us the number xy. (I know, I know. But trust me.) φ(x + y) is equal to log(xy), which equals log(x) + log(y), which is the same number as φ(x) + φ(y). There’s a way to see the postive real numbers being multiplied together as equivalent to all the real numbers being added together.

You might figure that the positive real numbers and all the real numbers aren’t very different-looking things. Perhaps so. Here’s another example I like, drawn from Wikipedia’s entry on Isomorphism. It has as sets things that don’t seem to have anything to do with one another.

Let me have another brand-new group G. It has as its set the whole numbers 0, 1, 2, 3, 4, and 5. Its operation is addition modulo 6. So 2 + 2 is 4, while 2 + 3 is 5, and 2 + 4 is 0, and 2 + 5 is 1, and so on. You get the pattern, I hope.

The brand-new group H, now, that has a more complicated-looking set. Its set is ordered pairs of whole numbers, which I’ll represent as (a, b). Here ‘a’ may be either 0 or 1. ‘b’ may be 0, 1, or 2. To describe its addition rule, let me say we have the elements (a, b) and (c, d). Find their sum first by adding together a and c, modulo 2. So 0 + 0 is 0, 1 + 0 is 1, 0 + 1 is 1, and 1 + 1 is 0. That result is the first number in the pair. The second number we find by adding together b and d, modulo 3. So 1 + 0 is 1, and 1 + 1 is 2, and 1 + 2 is 0, and so on.

So, for example, (0, 1) plus (1, 1) will be (1, 2). But (0, 1) plus (1, 2) will be (1, 0). (1, 2) plus (1, 0) will be (0, 2). (1, 2) plus (1, 2) will be (0, 1). And so on.

The isomorphism matches up things in G to things in H this way:

In G φ(G), in H
0 (0, 0)
1 (1, 1)
2 (0, 2)
3 (1, 0)
4 (0, 1)
5 (1, 2)

I recommend playing with this a while. Pick any pair of numbers x and y that you like from G. And check their matching ordered pairs φ(x) and φ(y) in H. φ(x + y) is the same thing as φ(x) + φ(y) even though the things in G’s set don’t look anything like the things in H’s.

Isomorphisms exist for other structures. The idea extends the way homomorphisms do. A ring, for example, has two operations which we think of as addition and multiplication. An isomorphism matches two rings in ways that preserve the addition and multiplication, and which match everything in the first ring’s set to everything in the second ring’s set, one-to-one. The idea of the isomorphism is that two different things can be paired up so that they look, and work, remarkably like one another.

One of the common uses of isomorphisms is describing the evolution of systems. We often like to look at how some physical system develops from different starting conditions. If you make a little variation in how things start, does this produce a small change in how it develops, or does it produce a big change? How big? And the description of how time changes the system is, often, an isomorphism.

Isomorphisms also appear when we study the structures of groups. They turn up naturally when we look at things called “normal subgroups”. The name alone gives you a good idea what a “subgroup” is. “Normal”, well, that’ll be another essay.

A Leap Day 2016 Mathematics A To Z: Homomorphism


I’m not sure how, but many of my Mathematics A To Z essays seem to circle around algebra. I mean abstract algebra, not the kind that involves petty concerns like ‘x’ and ‘y’. In abstract algebra we worry about letters like ‘g’ and ‘h’. For special purposes we might even have ‘e’. Maybe it’s that the subject has a lot of familiar-looking words. For today’s term, I’m doing an algebra term, and one that wasn’t requested. But it’ll make my life a little easier when I get to a word that was requested.

Homomorphism.

Also, I lied when I said this was an abstract algebra word. At least I was imprecise. The word appears in a fairly wide swath of mathematics. But abstract algebra is where most mathematics majors first encounter it. And the other uses hearken back to this. If you understand what an algebraist means by “homomorphism” then you understand the essence of what someone else means by it.

One of the things mathematicians study a lot is mapping. This is matching the things in one set to things in another set. Most often we want this to be done by some easy-to-understand rule. Why? Well, we often want to understand how one group of things relates to another group. So we set up maps between them. These describe how to match the things in one set to the things in another set. You may think this sounds like it’s just a function. You’re right. I suppose the name “mapping” carries connotations of transforming things into other things that a “function” might not have. And “functions”, I think, suggest we’re working with numbers. “Mappings” sound more abstract, at least to my ear. But it’s just a difference in dialect, not substance.

A homomorphism is a mapping that obeys a couple of rules. What they are depends on the kind of things the homomorphism maps between. I want a simple example, so I’m going to use groups.

A group is made up of two things. One is a set, a collection of elements. For example, take the whole numbers 0, 1, 2, and 3. That’s a good enough set. The second thing in the group is an operation, something to work like addition. For example, we might use “addition modulo 4”. In this scheme, addition (and subtraction) work like they do with ordinary whole numbers. But if the result would be more than 3, we subtract 4 from the result, until we get something that’s 0, 1, 2, or 3. Similarly if the result would be less than 0, we add 4, until we get something that’s 0, 1, 2, or 3. The result is an addition table that looks like this:

+ 0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2

So let me call G the group that has as its elements 0, 1, 2, and 3, and that has addition be this modulo-4 addition.

Now I want another group. I’m going to name it H, because the alternative is calling it G2 and subscripts are tedious to put on web pages. H will have a set with the elements 0, 1, 2, 3, 4, 5, 6, and 7. Its addition will be modulo-8 addition, which works the way you might have guessed after looking at the above. But here’s the addition table:

+ 0 1 2 3 4 5 6 7
0 0 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7 0
2 2 3 4 5 6 7 0 1
3 3 4 5 6 7 0 1 2
4 4 5 6 7 0 1 2 3
5 5 6 7 0 1 2 3 4
6 6 7 0 1 2 3 4 5
7 7 0 1 2 3 4 5 6

G and H look a fair bit like each other. Their sets are made up of familiar numbers, anyway. And the addition rules look a lot like what we’re used to.

We can imagine mapping from one to the other pretty easily. At least it’s easy to imagine mapping from G to H. Just match a number in G’s set — say, ‘1’ — to a number in H’s set — say, ‘2’. Easy enough. We’ll do something just as daring in matching ‘0’ to ‘1’, and we’ll map ‘2’ to ‘3’. And ‘3’? Let’s match that to ‘4’. Let me call that mapping f.

But f is not a homomorphism. What makes a homomorphism an interesting map is that the group’s original addition rule carries through. This is easier to show than to explain.

In the original group G, what’s 1 + 2? … 3. That’s easy to work out. But in H, what’s f(1) + f(2)? f(1) is 2, and f(2) is 3. So f(1) + f(2) is 5. But what is f(3)? We set that to be 4. So in this mapping, f(1) + f(2) is not equal to f(3). And so f is not a homomorphism.

Could anything be? After all, G and H have different sets, sets that aren’t even the same size. And they have different addition rules, even if the addition rules look like they should be related. Why should we expect it’s possible to match the things in group G to the things in group H?

Let me show you how they could be. I’m going to define a mapping φ. The letter’s often used for homomorphisms. φ matches things in G’s set to things in H’s set. φ(0) I choose to be 0. φ(1) I choose to be 2. φ(2) I choose to be 4. φ(3) I choose to be 6.

And now look at this … φ(1) + φ(2) is equal to 2 + 4, which is 6 … which is φ(3). Was I lucky? Try some more. φ(2) + φ(2) is 4 + 4, which in the group H is 0. In the group G, 2 + 2 is 0, and φ(0) is … 0. We’re all right so far.

One more. φ(3) + φ(3) is 6 + 6, which in group H is 4. In group G, 3 + 3 is 2. φ(2) is 4.

If you want to test the other thirteen possibilities go ahead. If you want to argue there’s actually only seven other possibilities do that, too. What makes φ a homomorphism is that if x and y are things from the set of G, then φ(x) + φ(y) equals φ(x + y). φ(x) + φ(y) uses the addition rule for group H. φ(x + y) uses the addition rule for group G. Some mappings keep the addition of things from breaking. We call this “preserving” addition.

This particular example is called a group homomorphism. That’s because it’s a homomorphism that starts with one group and ends with a group. There are other kinds of homomorphism. For example, a ring homomorphism is a homomorphism that maps a ring to a ring. A ring is like a group, but it has two operations. One works like addition and the other works like multiplication. A ring homomorphism preserves both the addition and the multiplication simultaneously.

And there are homomorphisms for other structures. What makes them homomorphisms is that they preserve whatever the important operations on the strutures are. That’s typically what you might expect when you are introduced to a homomorphism, whatever the field.