Up until my last post, I feel like the things I've talked about have flowed pretty linearly. However, math doesn't usually work that way. That often seems counter-intuitive because since first grade everything that we learn builds off of each previous topic, but usually math is all over the place and different topics influence each other in many different ways. This is usually why its so hard to read a book because books, by nature, are linear but math is not. Because of this, I'm going to step back away from isomorphisms and the symmetric groups and talk about cosets, which will eventually get us to normal subgroups and factor groups. There are some properties about cosets that are important to understand that I also need to introduce. These properties may not be obvious, so I'm going to prove them, but you are welcome to just trust me.
Definition: Left and Right Cosets.
Let H be a subgroup of a group, G and let a∈G. We then define a∙H = {a∙h : h∈H} and call it the left coset of H in G containing a and we define H∙a = {h∙a : h∈H} and call it the right coset of H in G containing a. In either case, a is called the coset representative of a∙H or H∙a.
Let H be a subgroup of a group, G and let a∈G. We then define a∙H = {a∙h : h∈H} and call it the left coset of H in G containing a and we define H∙a = {h∙a : h∈H} and call it the right coset of H in G containing a. In either case, a is called the coset representative of a∙H or H∙a.
It is noted in the definition that H is a subgroup of G, but even if H is only a subset of G, the notations of a∙H and H∙a are still valid, but they are not called cosets. It will be rare - if ever - that we use a∙H and H∙a when H is not a subgroup in this blog. This definition of a coset is very simple - all you do to calculate a coset, a∙H, is to take every element of H and multiply it on the left by a. What we are mostly concerned with is the interaction between between different cosets. First, though, I'm going to give an example where I calculate all of the left cosets of a subgroup of D4.
Example: Cosets of the Dihedral Group
Let G = D4 = {e,R90,R180,R270,Fh,Fr,Fv,Fl} and let H = {e,Fh}. It is very simple to verify that H is a subgroup. Below I have calculated each of the left cosets of H in G.
Let G = D4 = {e,R90,R180,R270,Fh,Fr,Fv,Fl} and let H = {e,Fh}. It is very simple to verify that H is a subgroup. Below I have calculated each of the left cosets of H in G.
| e∙H | = | {e,Fh} |
| R90∙H | = | {R90,Fr} |
| R180∙H | = | {R180,Fv} |
| R270∙H | = | {R270,Fl} |
| Fh∙H | = | {Fh,e} |
| Fr∙H | = | {Fr,R90} |
| Fv∙H | = | {Fv,R180} |
| Fl∙H | = | {Fl,R270} |
This is not meant to just demonstrate a routine calculation. You should look at these coset calculations and try to see patterns. There are a lot of things to notice, and those patterns are the properties that I'm going to prove. The first thing that you should notice, that isn't really a theorem, is that most of the cosets are not subgroups - in fact most of them don't even have the identy element. The only cosets that are subgroups of D4 are the ones that are equal to H.
Theorem:
Let G be a group and H a subgroup of G. Then a∙H = H if and only if a∈H.
Let G be a group and H a subgroup of G. Then a∙H = H if and only if a∈H.
Proof:
First, suppose that a∙H = H. Then a=a∙e∈a∙H=H. Next, assume that a∈H. Since H is closed, we get that a∙H⊆H. To show that H⊆a∙H, let h∈H. Note that since a∈H, a-1∈H, and since h∈H, a-1∙h∈H. Now we get that h=e∙h=(a∙a-1)∙h=a∙(a-1∙h)∈a∙H so H⊆a∙H and a∙H = H.
First, suppose that a∙H = H. Then a=a∙e∈a∙H=H. Next, assume that a∈H. Since H is closed, we get that a∙H⊆H. To show that H⊆a∙H, let h∈H. Note that since a∈H, a-1∈H, and since h∈H, a-1∙h∈H. Now we get that h=e∙h=(a∙a-1)∙h=a∙(a-1∙h)∈a∙H so H⊆a∙H and a∙H = H.
This property illuminates something very interesting. We noted before that some cosets of H in G from our example are actually equal to H. What this gives us is that this happens precisely when the coset representative is in H which is a very nice condition.
Theorem:
Let G be a group and H a subgroup of G. Then ∀a,b∈G, a∙H = b∙H if and only if a-1∙b∈H.
Let G be a group and H a subgroup of G. Then ∀a,b∈G, a∙H = b∙H if and only if a-1∙b∈H.
Proof:
We observe that a∙H = b∙H if and only if H = (a-1∙b)∙H and from the previous theorem, H = (a-1∙b)∙H if and only if a-1∙b∈H.
We observe that a∙H = b∙H if and only if H = (a-1∙b)∙H and from the previous theorem, H = (a-1∙b)∙H if and only if a-1∙b∈H.
This is a bit of an extension of our last property. An immediate consequence is that two cosets are equal when the representative of one lies in the other (that is a∙H = b∙H when a∈b∙H and b∈a∙H). This is pretty cool because it means that if we have a coset of H, then we can choose any element in that coset to be its representative. Which means that if K is some coset of H in G, then ∀a∈K, K = a∙H. This is very, very useful.
Theorem:
Let G be a group and H a subgroup of G. Then ∀a,b∈G, either a∙H = b∙H or a∙H∩b∙H = ∅.
Let G be a group and H a subgroup of G. Then ∀a,b∈G, either a∙H = b∙H or a∙H∩b∙H = ∅.
Proof:
Suppose ∃x∈a∙H∩b∙H. Then a-1∙x∈H so a∙H = x∙H and similarly b-1∙x∈H so b∙H = x∙H. Finally we have that a∙H = b∙H and so either a∙H = b∙H or a∙H∩b∙H = ∅.
Suppose ∃x∈a∙H∩b∙H. Then a-1∙x∈H so a∙H = x∙H and similarly b-1∙x∈H so b∙H = x∙H. Finally we have that a∙H = b∙H and so either a∙H = b∙H or a∙H∩b∙H = ∅.
The statement of this property might be a little bit confusing, but what it essentially means is that given two cosets, either they are the same or they have no elements in common. The last property told us how to know if two cosets are the same and this property tells us that if they are not the same, then they are completely distinct. This means that the set of cosets of a particular subgroup partitions G - or basically that if H is a subgroup of G, then every single element of G is in exactly one coset of H.
Theorem:
Let G be a group and H a subgroup of G. Then ∀a,b∈G, |a∙H| = |b∙H|.
Let G be a group and H a subgroup of G. Then ∀a,b∈G, |a∙H| = |b∙H|.
Proof:
Define the function, φ:a∙H→b∙H by φ(a∙h)=b∙h ∀a∙h∈a∙H. This is obviously a surjection, and it is an injection because cancellation gives that a∙h=b∙h implies that a=b. Since there exists a bijection between a∙H and b∙H, it then follows that |a∙H| = |b∙H|.
Define the function, φ:a∙H→b∙H by φ(a∙h)=b∙h ∀a∙h∈a∙H. This is obviously a surjection, and it is an injection because cancellation gives that a∙h=b∙h implies that a=b. Since there exists a bijection between a∙H and b∙H, it then follows that |a∙H| = |b∙H|.
Finally, we have that if H is a subgroup of G, then every coset of H is the exact same size. Now we already saw that the cosets of H partition G, but we now see that these cosets partition G into partitions of the exact same size.
Now we can try to put everything we've learned about cosets together. Suppose G is a group with subgroup H and suppose that K is a coset of H in G. We know that ∀a∈K, K = a∙H so that any element of K can be chosen as its coset representative. We also know that two different cosets are in fact completely disjoint - that is that they have no elements in common - but are the exact same size. You can (and should) go back up to my example of all the cosets of {e,Fh} in D4 and verify that these four properties hold. One final, very important thing to mention is that even though all of these theorems that I gave were concerned with left cosets, they all have analogous results for right cosets.
We learned a lot about cosets today, but I haven't yet explained how they're useful and it is definitely not obvious. As we'll see, if G is a group and N is a certain type of subgroup (called a normal subgroup), then the sets of all cosets of N in G form a group in itself, called a factor group, and these are extremely important in the world of group theory. Factor groups have a lot of very interesting, very important properties, and there are a lot of proofs in group theory that involve showing a property about a group by inspecting its factor groups, and a lot of these things could not be proved any other way.
References
Previous Related Post: Early Properties of Groups
Text Reference: Gallain Chapter 7
Wikipedia: Cosets
Wolfram Mathworld: Cosets
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