My 2018 Mathematics A To Z: Group Action


I got several great suggestions for topics for ‘g’. The one that most caught my imagination was mathtuition88’s, the group action. Mathtuition88 is run by Mr Wu, a mathematics tutor in Singapore. His mathematics blog recounts his own explorations of interesting topics.

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Group Action.

This starts from groups. A group, here, means a pair of things. The first thing is a set of elements. The second is some operation. It takes a pair of things in the set and matches it to something in the set. For example, try the integers as the set, with addition as the operation. There are many kinds of groups you can make. There can be finite groups, ones with as few as one element or as many as you like. (The one-element groups are so boring. We usually need at least two to have much to say about them.) There can be infinite groups, like the integers. There can be discrete groups, where there’s always some minimum distance between elements. There can be continuous groups, like the real numbers, where there’s no smallest distance between distinct elements.

Groups came about from looking at how numbers work. So the first examples anyone gets are based on numbers. The integers, especially, and then the integers modulo something. For example, there’s Z_2 , which has two numbers, 0 and 1. Addition works by the rule that 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, and 1 + 1 = 0. There’s similar rules for Z_3 , which has three numbers, 0, 1, and 2.

But after a few comfortable minutes on this, group theory moves on to more abstract things. Things with names like the “permutation group”. This starts with some set of things and we don’t even care what the things are. They can be numbers. They can be letters. They can be places. They can be anything. We don’t care. The group is all of the ways to swap elements around. All the relabellings we can do without losing or gaining an item. Or another, the “symmetry group”. This is, for some given thing — plates, blocks, and wallpaper patterns are great examples — all the ways you can rotate or move or reflect the thing without changing the way it looks.

And now we’re creeping up on what a “group action” is. Let me just talk about permutations here. These are where you swap around items. Like, start out with a list of items “1 2 3 4”. And pick out a permutation, say, swap the second with the fourth item. We write that, in shorthand, as (2 4). Maybe another permutation too. Say, swap the first item with the third. Write that out as (1 3). We can multiply these permutations together. Doing these permutations, in this order, has a particular effect: it swaps the second and fourth items, and swaps the first and third items. This is another permutation on these four items.

These permutations, these “swap this item with that” rules, are a group. The set for the group is instructions like “swap this with that”, or “swap this with that, and that with this other thing, and this other thing with the first thing”. Or even “leave this thing alone”. The operation between two things in the set is, do one and then the other. For example, (2 3) and then (3 4) has the effect of moving the second thing to the fourth spot, the (original) fourth thing to the third spot, and the original third thing to the second spot. That is, it’s the permutation (2 3 4). If you ever need something to doodle during a slow meeting, try working out all the ways you can shuffle around, say, six things. And what happens as you do all the possible combinations of these things. Hey, you’re only permuting six items. How many ways could that be?

So here’s what sounds like a fussy point. The group here is made up the ways you can permute these items. The items aren’t part of the group. They just gave us something to talk about. This is where I got so confused, as an undergraduate, working out groups and group actions.

When we move back to talking about the original items, then we get a group action. You get a group action by putting together a group with some set of things. Let me call the group ‘G’ and the set ‘X’. If I need something particular in the group I’ll call that ‘g’. If I need something particular from the set ‘X’ I’ll call that ‘x’. This is fairly standard mathematics notation. You see how subtly clever this notation is. The group action comes from taking things in G and applying them to things in X, to get things in X. Usually other things, but not always. In the lingo, we say the group action maps the pair of things G and X to the set X.

There are rules these actions have to follow. They’re what you would expect, if you’ve done any fiddling with groups. Don’t worry about them. What’s interesting is what we get from group actions.

First is group orbits. Take some ‘g’ out of the group G. Take some ‘x’ out of the set ‘X’. And build this new set. First, x. Then, whatever g does to x, which we write as ‘gx’. But ‘gx’ is still something in ‘X’, so … what does g do to that? So toss in ‘ggx’. Which is still something in ‘X’, so, toss in ‘gggx’. And ‘ggggx’. And keep going, until you stop getting new things. If ‘X’ is finite, this sequence has to be finite. It might be the whole set of X. It might be some subset of X. But if ‘X’ is finite, it’ll get back, eventually, to where you started, which is why we call this the “group orbit”. We use the same term even if X isn’t finite and we can’t guarantee that all these iterations of g on x eventually get back to the original x. This is a subgroup of X, based on the same group operation that G has.

There can be other special groups. Like, are there elements ‘g’ that map ‘x’ to ‘x’? Sure. The has to be at least one, since the group G has an identity element. There might be others. So, for any given ‘x’, what are all the elements in ‘g’ that don’t change it? The set of all the values of g for which gx is x is the “isotropy group” Gx. Or the “stabilizer subgroup”. This is a subgroup of G, based on x.

Yes, but the point?

Well, the biggest thing we get from group actions is the chance to put group theory principles to work on specific things. A group might describe the ways you can rotate or reflect a square plate without leaving an obvious change in the plate. The group action lets you make this about the plate. Much of modern physics is about learning how the geometry of a thing affects its behavior. This can be the obvious sorts of geometry, like, whether it’s rotationally symmetric. But it can be subtler things, like, whether the forces in the system are different at different times. Group actions let us put what we know from geometry and topology to work in specifics.

A particular favorite of mine is that they let us express the wallpaper groups. These are the ways we can use rotations and reflections and translations (linear displacements) to create different patterns. There are fewer different patterns than you might have guessed. (Different, here, overlooks such petty things as whether the repeated pattern is a diamond, a flower, or a hexagon. Or whether the pattern repeats every two inches versus every three inches.)

And they stay useful for abstract mathematical problems. All this talk about orbits and stabilizers lets us find something called the Orbit Stabilization Theorem. This connects the size of the group G to the size of orbits of x and of the stabilizer subgroups. This has the exciting advantage of letting us turn many proofs into counting arguments. A counting argument is just what you think: showing there’s as many of one thing as there are another. here’s a nice page about the Orbit Stabilization Theorem, and how to use it. This includes some nice, easy-to-understand problems like “how many different necklaces could you make with three red, two green, and one blue bead?” Or if that seems too mundane a problem, an equivalent one from organic chemistry: how many isomers of naphthol could there be? You see where these group actions give us useful information about specific problems.


If you should like a more detailed introduction, although one that supposes you’re more conversant with group theory than I do here, this is a good sequence: Group Actions I, which actually defines the things. Group actions II: the orbit-stabilizer theorem, which is about just what it says. Group actions III — what’s the point of them?, which has the sort of snappy title I like, but which gives points that make sense when you’re comfortable talking about quotient groups and isomorphisms and the like. And what I think is the last in the sequence, Group actions IV: intrinsic actions, which is about using group actions to prove stuff. And includes a mention of one of my favorite topics, the points the essay-writer just didn’t get the first time through. (And more; there’s a point where the essay goes wrong, and needs correction. I am not the Joseph who found the problem.)

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The End 2016 Mathematics A To Z: Monster Group


Today’s is one of my requested mathematics terms. This one comes to us from group theory, by way of Gaurish, and as ever I’m thankful for the prompt.

Monster Group.

It’s hard to learn from an example. Examples are great, and I wouldn’t try teaching anything subtle without one. Might not even try teaching the obvious without one. But a single example is dangerous. The learner has trouble telling what parts of the example are the general lesson to learn and what parts are just things that happen to be true for that case. Having several examples, of different kinds of things, saves the student. The thing in common to many different examples is the thing to retain.

The mathematics major learns group theory in Introduction To Not That Kind Of Algebra, MAT 351. A group extracts the barest essence of arithmetic: a bunch of things and the ability to add them together. So what’s an example? … Well, the integers do nicely. What’s another example? … Well, the integers modulo two, where the only things are 0 and 1 and we know 1 + 1 equals 0. What’s another example? … The integers modulo three, where the only things are 0 and 1 and 2 and we know 1 + 2 equals 0. How about another? … The integers modulo four? Modulo five?

All true. All, also, basically the same thing. The whole set of integers, or of real numbers, are different. But as finite groups, the integers modulo anything are nice easy to understand groups. They’re known as Cyclic Groups for reasons I’ll explain if asked. But all the Cyclic Groups are kind of the same.

So how about another example? And here we get some good ones. There’s the Permutation Groups. These are fun. You start off with a set of things. You can label them anything you like, but you’re daft if you don’t label them the counting numbers. So, say, the set of things 1, 2, 3, 4, 5. Start with them in that order. A permutation is the swapping of any pair of those things. So swapping, say, the second and fifth things to get the list 1, 5, 3, 4, 2. The collection of all the swaps you can make is the Permutation Group on this set of things. The things in the group are not 1, 2, 3, 4, 5. The things in the permutation group are “swap the second and fifth thing” or “swap the third and first thing” or “swap the fourth and the third thing”. You maybe feel uneasy about this. That’s all right. I suggest playing with this until you feel comfortable because it is a lot of fun to play with. Playing in this case mean writing out all the ways you can swap stuff, which you can always do as a string of swaps of exactly two things.

(Some people may remember an episode of Futurama that involved a brain-swapping machine. Or a body-swapping machine, if you prefer. The gimmick of the episode is that two people could only swap bodies/brains exactly one time. The problem was how to get everybody back in their correct bodies. It turns out to be possible to do, and one of the show’s writers did write a proof of it. It’s shown on-screen for a moment. Many fans were awestruck by an episode of the show inspiring a Mathematical Theorem. They’re overestimating how rare theorems are. But it is fun when real mathematics gets done as a side effect of telling a good joke. Anyway, the theorem fits well in group theory and the study of these permutation groups.)

So the student wanting examples of groups can get the Permutation Group on three elements. Or the Permutation Group on four elements. The Permutation Group on five elements. … You kind of see, this is certainly different from those Cyclic Groups. But they’re all kind of like each other.

An “Alternating Group” is one where all the elements in it are an even number of permutations. So, “swap the second and fifth things” would not be in an alternating group. But “swap the second and fifth things, and swap the fourth and second things” would be. And so the student needing examples can look at the Alternating Group on two elements. Or the Alternating Group on three elements. The Alternating Group on four elements. And so on. It’s slightly different from the Permutation Group. It’s certainly different from the Cyclic Group. But still, if you’ve mastered the Alternating Group on five elements you aren’t going to see the Alternating Group on six elements as all that different.

Cyclic Groups and Alternating Groups have some stuff in common. Permutation Groups not so much and I’m going to leave them in the above paragraph, waving, since they got me to the Alternating Groups I wanted.

One is that they’re finite. At least they can be. I like finite groups. I imagine students like them too. It’s nice having a mathematical thing you can write out in full and know you aren’t missing anything.

The second thing is that they are, or they can be, “simple groups”. That’s … a challenge to explain. This has to do with the structure of the group and the kinds of subgroup you can extract from it. It’s very very loosely and figuratively and do not try to pass this off at your thesis defense kind of like being a prime number. In fact, Cyclic Groups for a prime number of elements are simple groups. So are Alternating Groups on five or more elements.

So we get to wondering: what are the finite simple groups? Turns out they come in four main families. One family is the Cyclic Groups for a prime number of things. One family is the Alternating Groups on five or more things. One family is this collection called the Chevalley Groups. Those are mostly things about projections: the ways to map one set of coordinates into another. We don’t talk about them much in Introduction To Not That Kind Of Algebra. They’re too deep into Geometry for people learning Algebra. The last family is this collection called the Twisted Chevalley Groups, or the Steinberg Groups. And they .. uhm. Well, I never got far enough into Geometry I’m Guessing to understand what they’re for. I’m certain they’re quite useful to people working in the field of order-three automorphisms of the whatever exactly D4 is.

And that’s it. That’s all the families there are. If it’s a finite simple group then it’s one of these. … Unless it isn’t.

Because there are a couple of stragglers. There are a few finite simple groups that don’t fit in any of the four big families. And it really is only a few. I would have expected an infinite number of weird little cases that don’t belong to a family that looks similar. Instead, there are 26. (27 if you decide a particular one of the Steinberg Groups doesn’t really belong in that family. I’m not familiar enough with the case to have an opinion.) Funny number to have turn up. It took ten thousand pages to prove there were just the 26 special cases. I haven’t read them all. (I haven’t read any of the pages. But my Algebra professors at Rutgers were proud to mention their department’s work in tracking down all these cases.)

Some of these cases have some resemblance to one another. But not enough to see them as a family the way the Cyclic Groups are. We bundle all these together in a wastebasket taxon called “the sporadic groups”. The first five of them were worked out in the 1860s. The last of them was worked out in 1980, seven years after its existence was first suspected.

The sporadic groups all have weird sizes. The smallest one, known as M11 (for “Mathieu”, who found it and four of its siblings in the 1860s) has 7,920 things in it. They get enormous soon after that.

The biggest of the sporadic groups, and the last one described, is the Monster Group. It’s known as M. It has a lot of things in it. In particular it’s got 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 things in it. So, you know, it’s not like we’ve written out everything that’s in it. We’ve just got descriptions of how you would write out everything in it, if you wanted to try. And you can get a good argument going about what it means for a mathematical object to “exist”, or to be “created”. There are something like 1054 things in it. That’s something like a trillion times a trillion times the number of stars in the observable universe. Not just the stars in our galaxy, but all the stars in all the galaxies we could in principle ever see.

It’s one of the rare things for which “Brobdingnagian” is an understatement. Everything about it is mind-boggling, the sort of thing that staggers the imagination more than infinitely large things do. We don’t really think of infinitely large things; we just picture “something big”. A number like that one above is definite, and awesomely big. Just read off the digits of that number; it sounds like what we imagine infinity ought to be.

We can make a chart, called the “character table”, which describes how subsets of the group interact with one another. The character table for the Monster Group is 194 rows tall and 194 columns wide. The Monster Group can be represented as this, I am solemnly assured, logical and beautiful algebraic structure. It’s something like a polyhedron in rather more than three dimensions of space. In particular it needs 196,884 dimensions to show off its particular beauty. I am taking experts’ word for it. I can’t quite imagine more than 196,883 dimensions for a thing.

And it’s a thing full of mystery. This creature of group theory makes us think of the number 196,884. The same 196,884 turns up in number theory, the study of how integers are put together. It’s the first non-boring coefficient in a thing called the j-function. It’s not coincidence. This bit of number theory and this bit of group theory are bound together, but it took some years for anyone to quite understand why.

There are more mysteries. The character table has 194 rows and columns. Each column implies a function. Some of those functions are duplicated; there are 171 distinct ones. But some of the distinct ones it turns out you can find by adding together multiples of others. There are 163 distinct ones. 163 appears again in number theory, in the study of algebraic integers. These are, of course, not integers at all. They’re things that look like complex-valued numbers: some real number plus some (possibly other) real number times the square root of some specified negative number. They’ve got neat properties. Or weird ones.

You know how with integers there’s just one way to factor them? Like, fifteen is equal to three times five and no other set of prime numbers? Algebraic integers don’t work like that. There’s usually multiple ways to do that. There are exceptions, algebraic integers that still have unique factorings. They happen only for a few square roots of negative numbers. The biggest of those negative numbers? Minus 163.

I don’t know if this 163 appearance means something. As I understand the matter, neither does anybody else.

There is some link to the mathematics of string theory. That’s an interesting but controversial and hard-to-experiment-upon model for how the physics of the universe may work. But I don’t know string theory well enough to say what it is or how surprising this should be.

The Monster Group creates a monster essay. I suppose it couldn’t do otherwise. I suppose I can’t adequately describe all its sublime mystery. Dr Mark Ronan has written a fine web page describing much of the Monster Group and the history of our understanding of it. He also has written a book, Symmetry and the Monster, to explain all this in greater depths. I’ve not read the book. But I do mean to, now.

A Leap Day 2016 Mathematics A To Z: Normal Subgroup


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.

Normal Subgroup.

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.

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 Summer 2015 Mathematics A To Z: ring


Ring.

Early on in her undergraduate career a mathematics major will take a class called Algebra. Actually, Introduction to Algebra is more likely, but another Algebra will follow. She will have to explain to her friends and parents that no, it’s not more of that stuff they didn’t understand in high school about expanding binomial terms and finding quadratic equations. The class is the study of constructs that work much like numbers do, but that aren’t necessarily numbers.

The first structure studied is the group. That’s made of two components. One is a set of elements. There might be infinitely many of them — the real numbers, say, or the whole numbers. Or there might be finitely many — the whole numbers from 0 up to 11, or even just the numbers 0 and 1. The other component is an operation that works like addition. What we mean by “works like addition” is that you can take two of the things in the set, “add” them together, and get something else that’s in the set. It has to be associative: something plus the sum of two other things has to equal the sum of the first two things plus the third thing. That is, 1 + (2 + 3) is the same as (1 + 2) + 3.

Also, by the rules of what makes a group, the addition has to commute. First thing plus second thing has to be the same as second thing plus first thing. That is, 1 + 2 has the same value as 2 + 1 does. Furthermore, there has to be something called the additive identity. It works like zero does in ordinary arithmetic. Anything plus the additive identity is that original thing again. And finally, everything in the group has something that’s its additive inverse. The thing plus the additive inverse is the additive identity, our zero.

If you’re lost, that’s all right. A mathematics major spends as much as four weeks in Intro to Algebra feeling lost here. But this is an example. Suppose we have a group made up of the elements 0, 1, 2, and 3. 0 will be the additive identity: 0 plus anything is that original thing. So 1 plus 0 is 1. 1 plus 1 is 2. 1 plus 2 will be 3. 1 plus 3 will be … well, make that 0 again. 2 plus 0 is 2. 2 plus 1 will be 3. 2 plus 2 will be 0. 2 plus 3 will be 1. 3 plus 0 will be 3. 3 plus 1 will be 0. 3 plus 2 will be 1. 3 plus 3 will be 2. Plus will look like a very strange word at this point.

All the elements in this have an additive inverse. Add 3 to 1 and you get 0. Add 2 to 2 and you get 0. Add 1 to 3 and you get 0. And, yes, add 0 to 0 and you get 0. This means you get to do subtraction just as well as you get to do addition.

We’re halfway there. A “ring”, introduced just as the mathematics major has got the hang of groups, is a group with a second operation. Besides being a collection of elements and an addition-like operation, a ring also has a multiplication-like operation. It doesn’t have to do much, as a multiplication. It has to be associative. That is, something times the product of two other things has to be the same as the product of the first two things times the third. You’ve seen that, though. 1 x (2 x 3) is the same as (1 x 2) x 3. And it has to distribute: something times the sum of two other things has to be the same as the sum of the something times the first thing and the something times the second. That is, 2 x (3 + 4) is the same as 2 x 3 plus 2 x 4.

For example, the group we had before, 0 times anything will be 0. 1 times anything will be what we started with: 1 times 0 is 0, 1 times 1 is 1, 1 times 2 is 2, and 1 times 3 is 3. 2 times 0 is 0, 2 times 1 is 2, 2 times 2 will be 0 again, and 2 times 3 will be 2 again. 3 times 0 is 0, 3 times 1 is 3, 3 times 2 is 2, and 3 times 3 is 1. Believe it or not, this all works out. And “times” doesn’t get to look nearly so weird as “plus” does.

And that’s all you need: a collection of things, an operation that looks a bit like addition, and an operation that looks even more vaguely like multiplication.

Now the controversy. How much does something have to look like multiplication? Some people insist that a ring has to have a multiplicative identity, something that works like 1. The ring I described has one, but one could imagine a ring that hasn’t, such as the even numbers and ordinary addition and multiplication. People who want rings to have multiplicative identity sometimes use “rng” to speak — well, write — of rings that haven’t.

Some people want rings to have multiplicative inverses. That is, anything except zero has something you can multiply it by to get 1. The little ring I built there hasn’t got one, because there’s nothing you can multiply 2 by to get 1. Some insist on multiplication commuting, that 2 times 3 equals 3 times 2.

Who’s right? It depends what you want to do. Everybody agrees that a ring has to have elements, and addition, and multiplication, and that the multiplication has to distribute across addition. The rest depends on the author, and the tradition the author works in. Mathematical constructs are things humans find interesting to study. The details of how they’re made will depend on what work we want to do.

If a mathematician wishes to make clear that she expects a ring to have multiplication that commutes and to have a multiplicative identity she can say so. She would write that something is a commutative ring with identity. Or the context may make things clear. If you’re not sure, then you can suppose she uses the definition of “ring” that was in the textbook from her Intro to Algebra class sophomore year.

It may seem strange to think that mathematicians don’t all agree on what a ring is. After all, don’t mathematicians deal in universal, eternal truths? … And they do; things that are proven by rigorous deduction are inarguably true. But the parts of these truths that are interesting are a matter of human judgement. We choose the bunches of ideas that are convenient to work with, and give names to those. That’s much of what makes this glossary an interesting project.