Last time I gave a brief introduction to groups and a bit of motivation of group theory (symmetry). I suppose the next logical thing to do would be to introduce the notion of subgroups. However, there’s not terribly much to say on subgroups at this time, so I’ll say what they are, then proceed to examples and more examples of groups in general, and then talk about group homomorphisms.
It should be relatively obvious what type of concept is coming here. Just as a subset is a “set within a set,” a subgroup is a “group within a group.” That is,
Definition: Let be a group. If is a subset of , and is a group under the operation of , then is a subgroup of .
This is fairly straightforward, the only thing we should notice is that the subset has to be a group under the same operation as to be a subgroup. For example, we know that is a group under addition. It can be verified without much trouble that is also a group under addition, as associativity of addition is already satisfied, , and for any (so we have an identity), and whenever , we also have (if is even, so is ). Since , is a subgroup of . Similarly, for any , we have as a subgroup of . However, is it also illustrative to look at a non-example: let be the group of rational numbers under addition (if it’s not obvious, you should verify that this is indeed a group). We can see that is a group under multiplication (again, you should be able to verify this without much trouble); however, even though , under multiplication is not a subgroup of under addition, because they do not use the same operation. I’m not going to spend too much time on examples here; we’ll see a few in a moment when we look at homomorphisms.
Exercise: Let be a group, and let be subgroups of $G$. Show that the intersection is a subgroup of .
Mathematicians are very interested in the structure of objects. One might even go so far as to say that mathematics is the study of structure. What do I mean by “structure”? I mean the way something “looks” mathematically. The best way to do this is to look at an example: let’s look at two sets, and , and let’s suppose we equip with the binary operation and with the binary operation . We can describe these operations completely by just listing how they affect each of the combinations of elements in and , say
We can also write these in tables, which makes the structure of the binary operations a bit more apparent:
(I made you MS Paint tables because I don’t know how to format tables in WordPress. I’ll edit it if I ever figure out how to do so…) Now we see that if we swap the c’s and d’s in the second table, we have two tables that are basically the same, except the letters are different. In mathematics, we don’t really care too much about the names of the objects, just the way they behave. Here, the binary structures behave the same way, and we can create a bijective function between them that preserves the way they behave: take
We see that this mapping preserves the structures of the two sets, in that for any . We also don’t lose information, because is a bijection. As it turns out, this first condition is exactly what we want to describe functions betwixt groups. Without the structure-preserving condition above, the functions between groups would fail to give us very much information about the structure of the groups in question. So now we’ll define a group homomorphism with that in mind:
Definition: Let and be two groups with operations and , respectively. A function is called a (group) homomorphism if we have
for all .
Notice that the operation on the left side of the equation happens in , while the operation on the right side happens in . Also note that we relaxed the condition that be a bijection. If it is, then we call a (group) isomorphism. In the case of a homomorphism, we can lost some of the information of , whereas in the case of an isomorphism, we essentially retain all of the information about the original group. Let’s look at some properties and examples to get a feel for these creatures.
Property: If is a homomorphism of groups, and is the identity of , then is the identity of .
Intuitively, this makes sense. If we’re going to respect the group structure, we should be sending the identity to the identity. We can also give a formal proof of this: , and since we’re in a group, we can multiply by the inverse of on the right on both sides of the equation (remember, multiplication on the right and left are two different things in general!) to see , but this simplifies to , which is what we wanted. In a similar manner, one can show
Property: If is a homomorphism of groups, and is any element of and is the inverse of , then , where is the inverse of .
What this is saying is that homomorphisms also respect inverses, which makes sense, again thinking that we defined them to be a structure-preserving map.
Here is also a good time to introduce the convention of “powers” in groups. It means exactly what you might think: , where we have ‘s on the right hand side. However, this isn’t how we define it formally, because when mathematicians want to be super precise, they
abhor don’t like ‘s.
Definition: Let be a group (written multiplicatively), let , and let . Then we define as follows:
- , where is the identity of ,
- If , , and
- If , then , where is the inverse of .
These powers work the way we want them to; i.e. , which also shows that powers of commute. Now, we’ll generally write arbitrary groups in multiplicative notation, but when working with commutative (AKA abelian) groups, we will usually write them additively. In this case, we won’t write , but rather . One has to be careful: we’re not actually multiplying by . This might not even make sense, as might not be a subset of the real numbers at all. However, even when it is, we’re technically not multiplying the element of the group by . (For the most part, if you do multiply by , you’ll get the right answer, but there are strange times when it won’t turn out right.)
Another property you might now suspect is
Property: If is a homomorphism of groups, and , then .
This can be proven via induction on .
Example: First, we’ll have to define a new group: let and . We can make this set into a group under addition, but if we try to add these normally, we see that isn’t closed. So we have to make a small modification. You’ll note that this set has all the remainders one can get upon division by , so when we add up two numbers and get a number larger than , we’ll just “wrap around” by defining an equivalence relation that relates each number to its remainder upon division by . This is the same equivalence relation I talked about in my post on basic set theory, except I used 5 rather than an arbitrary natural number. Formally, if , where means divides (recall that if and only if there exists some such that ). Then is a set of equivalence class representatives, and we’ll define addition on to be normal addition, and when we wind up with a number outside of the set of representatives, we simply take the number inside that is equivalent to the sum. If this seems confusing, don’t panic – just think about clocks! Whenever you need to calculate times, you work mod 12 (or mod 24 if you use 24 hour time). The only difference: when doing “clock arithmetic,” you use 12 as the equivalence class representative instead of 0. For example, : 11 o’clock plus 3 hours gives 2 o’clock. Under addition modulo , becomes a group. We call the “cyclic group of order .”
Now, we can define a homomorphism (actually we can define several). For the sake of concreteness, let’s use . First, let’s observe that any integer is the sum of some amount of 1′s or -1′s. What this means is that we can just take the element 1 in and add it to itself over and over (or add its inverse) to get any element of . So since a homomorphism has to respect the group operation, we only need to specify where 1 is sent under . We’ll denote the homomorphism that sends by : after that, it’s clear sailing! Let’s look at a few :
- : This homomorphism actually takes EVERYTHING in to 0, as we can see by writing (remember, this isn’t really multiplication, it’s operating by 1 times) so that . This homomorphism is the trivial homomorphism: we can always send everything in one group to the identity of another group. However, this homomorphism doesn’t give us any new information about the original group, so we focus on more interesting ones.
- : This homomorphism takes any member of and sends it to its equivalence class representative (where equivalence is modulo 12) in , which is the same as saying that the fiber over any element of is the equivalence class of that element under modulo 12 arithmetic. There’s some major collapsing going on for , but all of is hit.
- : Here, we only hit half of : we see , , , , , , and . That gives us all the information we need to know about this homomorphism.
I could keep going, but there’s not much new that happens after this. For other , we basically see the same phenomena. If we look at how much of is hit by , we see that hits 1 element, hits 12 elements, hits 6 elements, hits 4 elements, hits three elements, hits 12 elements, hits 2 elements, hits 12 elements… do we see the pattern? That’s right, the homomorphism hits elements. There are other patterns as well, two of which I am going to talk about next, and others that I will discuss in another post on cyclic groups.
A Taste of Some Special Subgroups Related to Homomorphisms
So far, I haven’t talked too much about subgroups. However, there are a few important subgroups that come up when we talk about homomorphisms. Before I give them away, go back and look at the homomorphisms above and play with them for a bit. Try finding some homomorphisms for various , or other homomorphisms in general. Once you’re done (or now, if you’re too lazy to experiment yourself) read on, fearless champion.
Image: Recall that the image of a function is defined as . As it turns out, the image of a group homomorphism is always a subgroup of . This shouldn’t be a surprise: when we make a homomorphism, we’re trying to preserve the group structure, and it wouldn’t make much sense if we were preserving structure while somehow mapping to something that isn’t a group. It isn’t hard to check that in each above, the image is a subgroup. It would be good of you to try to prove such a thing.
Kernel: The kernel of a group homomorphism is possibly even more important than the idea of the image of a group homomorphism. We’ll begin with the definition:
Definition: Let be a group homomorphism. Then the kernel of is defined to be
where is the identity of .
So the kernel of a group homomorphism is the set of elements in the domain that map to the identity element in the image, or the fiber over the identity in the image. This is also easily seen to be a subgroup (exercise!) and it relates homomorphisms of groups to other objects (that we will talk about soon) and gives us information about when a group homomorphism is injective.
Example: Consider the map . What is the kernel? Well, it’s the elements that map to the identity, 0. So what maps to 0? Obviously, . We can also see that since , we have . Noting that exactly when for some , we see that (and this is certainly a subgroup of ).
So that’s it for now! Next time, I may or may not discuss more things about kernels. I also may or may not talk about other just straight up important concepts. Please accept my apologies for the wait on this post, and I’ll see you then!