###### (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, $G$, is a set with a binary operation $\mu: G \times G \to G$ which is associative, has an identity, and has inverse elements.

## Example: $\mathbb{Z} / 4 \mathbb{Z}$

We’re going to next consider how to take a group, $G$ and a subgroup, $H$ and form the quotient group $G/H$.  As an example, let’s use the group $(\mathbb{Z}, +)$ of integers under addition.  Consider the subgroup $4 \mathbb{Z}$, which contains elements like 12, 4, -100, etc, but not 3, 2, -99, etc.  We will now define an equivalence relation on $\mathbb{Z}$ by saying that the elements $x$ and $y$ are equivalent, which we will denote by $x \sim y$, if $x - y \in 4\mathbb{Z}$.

Let’s go through some examples of elements which are equivalent:

• $4 - 0 = 4 \in 4 \mathbb{Z}$ and so $4 \sim 0$. [Edited 6/13]
• $4 - 8 = -4 \in 4 \mathbb{Z}$ and so $4 \sim -8$.
• $4 - 5 = -1 \notin 4 \mathbb{Z}$ and so $4 \not\sim 5$.

If we went through all possible combinations, we would see that there are exactly 4 equivalence classes of elements that we are partitioning $\mathbb{Z}$ into:

• $4 \mathbb{Z}+ 0 = \{\ldots, -12, -8, -4, 0, 4, 8, \ldots \}$
• $4 \mathbb{Z}+ 1 = \{\ldots, -11, -7, -3, 1, 5, 9, \ldots \}$
• $4 \mathbb{Z}+ 2 = \{\ldots, -10, -6, -2, 2, 6, 10, \ldots \}$
• $4 \mathbb{Z}+ 3 = \{\ldots, -9, -5, -1, 3, 7, 11, \ldots \}$

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, $\mathbb{Z}$, and just treat it as four elements: $[0] = 4 \mathbb{Z}+ 0$$[1] = 4 \mathbb{Z}+ 1$$[2] = 4 \mathbb{Z}+ 2$, and $[3] = 4 \mathbb{Z}+ 3$.  Note: a shortcut for knowing which set is which is by looking at the remainders of the elements, so 5 is in $[1]$ 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 $4 \mathbb{Z}$!

So now let’s add elements in our set, $\mathbb{Z} / 4 \mathbb{Z} = \{ [0], [1], [2], [3] \}$.  Remember that each $[x]$ is a subset of integers.  Suppose I took the number $2 \in [2]$ and added it to the element $3 \in [3]$.  Well 2+3 = 5, but there is no element “$[5]$“, is there?  Well if there was, what would it be?  The set $[5]$ would be the set of integers which are equivalent to the number 5, under the rule “~” we used above.  Well note that $5 \in [1]$, and so it turns out that $[5]=[1]$ (by the properties of an equivalence relation, which we will have to go through carefully and formally).  The point is… $[2] + [3] = [5] =[1]$!!

## Example: $\mathbb{Z} / 5 \mathbb{Z}$

Let us do the same thing as we did above to partition the integers, except now $x \sim y$ if $x -y \in 5 \mathbb{Z}$.  Note that we now have 5 elements in our set, which the rest of the world refers to as $\mathbb{Z}_5 := \{ [1], [2], [3], [4], [5]\}$.  Note that I chose to use $[5]$ in place of $[0]$; 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: $0 \to \mathbb{Z} \to \mathbb{R} \to S^1 \to 0$

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, $\mathbb{Z}$, which are a group under addition.  But now consider the subgroup $2 \pi \mathbb{Z} \subset \mathbb{R}$ of integer-spaced multiples of $2 \pi$, each of which is a real number (whatever that means, amiright?).  Anyway, I’m now going to consider the group $\mathbb{R} / 2 \pi \mathbb{Z}$, but what the heck is this group?

Well I chose $2 \pi$ because that is the circumference of the unit circle.  So notice I have a function $f: \mathbb{R} \to \mathbb{R}^2$ by sending $t \mapsto (\cos (t), \sin (t))$ which takes each real number and maps it on the unit circle.  Which real numbers go to zero?  The multiples of $2 \pi$!  Which real numbers get sent to $\frac{\pi}{3}$?  The real numbers $2 \pi \cdot n + \frac{\pi}{3}$ where $n$ is any integer!

So we’ve taken the real line, and thought of any two numbers which are $2\pi$-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 $5 \mathbb{Z}$, 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, $S^1$.  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.

## 2 thoughts on “Quotients of Groups: Where 2+3 = 1.”

1. Joshua Papello says:

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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