###### (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.

## Example:

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 [1] 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… !!

## Example:

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.

## Example:

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*.

Thanks for the content, but I think you have a typo! Where you specify examples of which two elements are equivalent, your third example shows that 4 is clearly not equivalent to 5, but your first example shows that 4 is equivalent to 5…! I noticed it as soon as I realized that 5 is clearly not a multiple of 4, so maybe you meant to say that 4 ~ 4.

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Thanks, Josh! I’m fixing this now!

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