(Meant for my Intro to Homological Algebra Summer Seminar Group at SJC)
Last time we met we discussed groups, where we can recall that
A group, , is a set with a binary operation which is associative, has an identity, and has inverse elements.
We’re going to next consider how to take a group, and a subgroup, and form the quotient group . As an example, let’s use the group of integers under addition. Consider the subgroup , which contains elements like 12, 4, -100, etc, but not 3, 2, -99, etc. We will now define an equivalence relation on by saying that the elements and are equivalent, which we will denote by , if .
Let’s go through some examples of elements which are equivalent:
- and so . [Edited 6/13]
- and so .
- and so .
If we went through all possible combinations, we would see that there are exactly 4 equivalence classes of elements that we are partitioning into:
Just to absolutely clear, for each of these sets, if I take two elements in the same set, and I look at their difference, it will be a multiple of 4; if I take two elements each from different sets, their difference will not be a multiple of 4.
Here’s where things get trippy: we will now take the infinitely-large set, , and just treat it as four elements: , , , and . Note: a shortcut for knowing which set is which is by looking at the remainders of the elements, so 5 is in because when I divide it by 4, the remainder is 1. And -3 is in the  as well because when I consider: (-3) – (5)= – 8, I get an element in !
So now let’s add elements in our set, . Remember that each is a subset of integers. Suppose I took the number and added it to the element . Well 2+3 = 5, but there is no element ““, is there? Well if there was, what would it be? The set would be the set of integers which are equivalent to the number 5, under the rule “~” we used above. Well note that , and so it turns out that (by the properties of an equivalence relation, which we will have to go through carefully and formally). The point is… !!
Let us do the same thing as we did above to partition the integers, except now if . Note that we now have 5 elements in our set, which the rest of the world refers to as . Note that I chose to use in place of ; there is a reason which you might come to realize but it isn’t super important.
While the integers only form a group under addition, you should explore this new set and see if it forms a group under addition or multiplication or both!? Either way the answer should be surprising and interesting.
Finally, I would like to introduce a more geometric example. The previous examples ended up with a finite number of elements in our quotient, and everything was down to arithmetic. Consider again our lovely set of integers, , which are a group under addition. But now consider the subgroup of integer-spaced multiples of , each of which is a real number (whatever that means, amiright?). Anyway, I’m now going to consider the group , but what the heck is this group?
Well I chose because that is the circumference of the unit circle. So notice I have a function by sending which takes each real number and maps it on the unit circle. Which real numbers go to zero? The multiples of ! Which real numbers get sent to ? The real numbers where is any integer!
So we’ve taken the real line, and thought of any two numbers which are -apart as being glued together, and if you try to draw this picture you should see a circle!
Well this picture isn’t really what I want you to see but I’m tired and it should be good enough to get your creative brain going!
Food For Thought: Preparing for our next meeting
After having slowly read through this blog post, which could take hours, I encourage you to try and go back through our notes to see if any of the examples, propositions, exercises, etc, make any more sense than they did. Specifically, I want us to finish up this discussion on quotients in our next meeting and then continue on to some other ideas, which I will now appropriately tease…
Note that we started with the additive group of integers, and when modding out by , we actually get some nice multiplicative structure. What if a group had both an additive and a multiplicative structure, do quotients make sense for playing nicely with both of these structures simultaneously? The answer to this leads us to a study of rings (groups which also have some multiplicative structure; not necessarily inverses), and modules (groups which can be multiplied by external elements; like vectors can be multiplied by numbers).
Finally, in the last example I gave, the title for that example had some arrows connecting the integers, to the real numbers, and then to the circle, . Those arrows are actually functions which allow the group structures of each group to communicate with each other. This type of function is called a homomorphism and will be central to our study of Homological Algebra.