A sequence of groups is just a list of groups with homomorphisms going down the list: . We’ll use to refer to the homomorphism from to . We say that a sequence is “exact” if the image of is the kernel of for each .
What does this mean? First off, if we start with an element of and hit it with we get something in . Since this is the same as , if we now apply we go to the identity element of . That is, the composition is always the trivial homomorphism sending all of to the identity in .
On the other hand, if we have an element in the kernel of it’s also in the image of . This means that if an element gets sent to the identity in the next group, it must have come from an element in the previous group.
So why should we care? Well, a number of different things can be very nicely said with exact sequences. If we write for the group containing only one element, we can set up a sequence: . What does it mean for this sequence to be exact? Well there’s only one homomorphism from to any group, and its image is just the identity element in . So the kernel of is trivial — is a monomorphism.
Now let’s flip the diagram over to . There’s only one homomorphism possible from any group to , and its kernel is the whole domain. This means that the image of has to be all of — is an epimorphism.
Let’s put these two together to get a sequence . Exactness at means that is a monomorphism, which we can think of as describing a copy of sitting inside . Exactness at means that is an epimorphism. What does exactness at mean? The image of is that copy of , which has to also be the kernel of . That is, is (isomorphic to) . We call any sequence of this form a “short exact sequence”.
Remember that the First Isomorphism Theorem tells us that we can factor any homomorphism into an epimorphism from the domain onto a quotient, followed by a monomorphism putting that quotient into the codomain. We can use that here to weave any exact sequence out of short exact sequences. Here is the (really cool) diagram:
Each of the diagonal lines is a short exact sequence, and as it says each (nontrivial) group off the main line is the image of one of the homomorphisms on the line and the kernel of the next.
We can also write an exact sequence . This just says that the homomorphism between and is an isomorphism. It’s really nice when this shows up in the middle of a longer exact sequence. If we can show that and are both trivial the sequence looks like , so and are immediately isomorphic.
Another way exact sequences show up is in describing the structure of a group. We know that every group is a quotient of a free group. That is, there is some free group so that is exact. Then the kernel of this projection is another group, so it’s the quotient of another free group . Now the sequence is exact. This is the presentation of by generators and relations. But the homomorphism from to might have a nontrivial kernel — there might be relations between the relations. In that case we can describe those relations as the quotent of another free group : is exact. We can keep going like this to construct an exact sequence called a “free resolution of “. It’s particularly nice if the process terminates at some point, giving a sequence . A free resolution of a group that has only finitely many terms gives a lot of information about the structure of .
Okay, now we’ve got all the setup for one big use of group theory.
Most mathematical structures come with some notion of symmetries. We can rotate a regular -sided polygon of a turn, or we can flip it over. We can rearrange the elements of a set. We can apply an automorphism of a group. The common thread in all these is a collection of reversible transformations. Performing one transformation and then the other is certainly also a transformation, so we can use this as a composition. The symmetries of a given structure form a group!
What if we paint one side of the above -gon black and the other white, so flipping it over definitely changes it. Then we can only rotate. The rotations are the collection of symmetries that preserve the extra structure of which side is which, and they form a subgroup of the group of all symmetries of the -gon. The symmetries of a structure preserving some extra structure form a subgroup!
As far as we’re concerned right now, mathematical structures are all built on sets. So the fundamental notion of symmetry is rearranging the elements of a set. Given a set , the set of all bijections from to itself is a group. We’ve actually seen a lot of these before. If is a finite set with elements, is the symmetric group on letters.
Now, a group action on a set is simply this: a homomorphism from to . That is, for each element in there is a bijection of and .
It’s important to note what seems like a switch here. It’s really due to the notation, but can easily seem confusing. What’s written on the right is “do the permutation corresponding to , then the one corresponding to “. So we have to think of the multiplication in as “first do , then do “.
In what follows I’ll often write as . The homomorphism property then reads
I’ll throw out a few definitions now, and follow up with examples in later posts.
We can slice up into subsets so that if and are in the same subset, for some , and not if they’re not in the same subset. In fact, this is rather like how we sliced up itself into cosets of . We call these slices the “orbits” of the action.
As an important special case of the principle that fixing additional structure induces subgroups, consider the “extra structure” of one identified point. We’re given an element of , and want to consider those transformations in which send to itself. Verify that this forms a subgroup of . We call it the “isotropy group” or the “stabilizer” of , and write it .
I’ll leave you to ponder this: if and are in the same -orbit, show that and are isomorphic.
Today I want to walk through what’s called the “First Isomorphism Theorem” for groups. There are two more, but the first is really more interesting in my view. I’ll start with a high-level sketch: kernels of homomorphisms are normal subgroups, images are quotient groups, and every homomorphism is a quotient followed by an isomorphism.
First I’m going to need a couple homomorphisms. If we’ve got a group and a normal subgroup , there’s immediately a homomorphism . Just send each to its coset . It should be clear that every coset gets hit at least once, so this is an epimorphism, and that its kernel is exactly . We call the “canonical projection” or the “canonical epimorphism” from to .
On the other hand, if is a group and is any subgroup, we have a homomorphism given by just sending every element of to itself inside . This is such a natural identification to make that it feels a little weird to think of it as a homomorphism at all, but it actually turns out to be quite useful. The kernel of is trivial, making it a monomorphism. We call it the “canonical injection” or “canonical monomorphism” from to .
Now consider any homomorphism . If is in the kernel of — is the identity of — and is any element of , we calculate
so is in the kernel as well. Thus the kernel is a normal subgroup.
So every kernel is a normal subgroup, and the canonical projection shows that every normal subgroup shows up as the kernel of some homomorphism.
Now we can write any homomorphism as a composition
where I’ve written the name of each composition next to its arrow. That is, we first project onto the quotient of the domain by the kernel of , then we send that to the image of by a homomorphism we call , and finally we inject the image into the codomain. As a bonus, is an isomorphism!
Okay, so how do we define ? If we write the kernel of as , we need to figure out what to do with a coset . If and are two elements of , then , so sends every element of to the same element of . We define .
Now let’s say . This means that , so is in already, and is the identity of . The kernel of is trivial, so is a monomorphism. On the other hand, every element in is for some , so each one is also for some . That makes an epimorphism, and thus an isomorphism. Q.E.D.
Every homomorphism works like this: you divide out some kernel, hit the quotient group with an isomorphism, and include the result into the target group. Since isomorphisms don’t really change anything about a group and the inclusion is pretty simple too, all the really interesting stuff goes on in the first step. The homomorphisms that can come out of are essentially determined by the normal subgroups of . Because of this, we call a group with no nontrivial normal subgroups “simple”. The kernel of an homomorphism from a simple group is either trivial or the whole group.
What we’re starting to see here is the tip of a much deeper approach to algebra. The internal structure of a group is intimately bound up with the external structure of the homomorphisms linking it to other groups. Each one determines, and is determined by the other, and this duality can be a powerful tool for answering questions on one side by turning them into questions on the other side.
Okay, it’s been pointed out to me that what I was thinking of in my update to yesterday’s post was a little more general than group theory. In the case of groups, preserving the composition is all that’s required. If I talk about semigroups later that extra condition will be needed.
So, I’ll leave it as a (relatively straightforward) exercise to show that if a function from one group to another preserves the composition that it also preserves identities. Oh, and inverses. May as well nail that down while we’re at it.
For now I’ll make this point, which I also make to all my calculus classes: it’s really not that bad to misremember something. That’s one of the nice things about mathematics. If you make a mistake it’s usually not hard to check. Also, when two people have different views on something it’s possible to check which one is right. There are real answers to be found. Despite the accusations of coldness and impersonality, it’s comforting to know that something has a real answer.
At last we come to the notion of a homomorphism of groups. These are really, in my view, the most important parts of the theory. They show up everywhere, and the structure of group theory is intimately bound up with the way homomorphisms work.
So what is a homomorphism? It’s a function from the set of members of one group to the set of members of another that “preserves the composition”. That is, a homomorphism takes an element of and gives back an element of . It has the further property that . The product of and uses the composition from , while the product of and uses the composition of .
Let’s consider an example very explicitly: a homomorphism . Remember that is the group of rearrangements of 3 objects (I’ll use a, b, and c), while is the group of “addition modulo 2”.
If we consider the permutations and in , each one is sent to 1 in the group , and 1+1 = 0 there. On the other hand, , which is sent to 0. The composition of the images is the image of the composition. We can pick any two permutations on the right and see the same thing.
Another example: with . The homomorphism property says that , and indeed we see that .
Another: . By . I mean the real numbers with addition as composition, and by . I mean the positive.nonzero real numbers with multiplication. I define . The laws of exponents tell us that .
As we continue we will see many more examples of homomorphisms. For now, there are a few definitions we will find useful later. Recall from the discussion about functions that a surjection is a function between functions that hits every point in its codomain at least once. A group homomorphism that is also a surjection we call an “epimorphism”. Similarly, an injection is a function that hits every point in its codomain at most once. A group homomorphism that is also an injection we call a “monomorphism”. A homomorphism that is both — the function is a bijection — we call an “isomorphism”. In the above examples, is an epimorphism, is a monomorphism, and is an isomorphism.
If a homomorphism’s domain and codomain group are the same, as in above, we call it an “endomorphism” on the group. If it’s also an isomorphism we call it an “automorphism”. The homomorphism is not an automorphism, since it doesn’t hit any point that’s not a multiple of 3.
And finally, a few things to think about.
- Can you construct a homomorphism from to similar to above, but for other values of ?
- What homomorphisms can you construct from to ? to ? to an arbitrary group ?
- What homomorphisms can you construct from to ?
UPDATE: I just remembered that I left off another technical requirement. A homomorphism has to send the identity of the first group to the identity of the second. It usually doesn’t cause a problem, but I should include it to be thorough. It isn’t hard to verify that all the homomorphisms I mentioned satisfy this property too.