and “Group 2.” How would you compare these two groups? That is, how would you determine

how similar or different they are from each other? And what do we mean by “similar”?

What features of groups are we even using to compare them? To answer these questions

and more, we will use a tool called a “homomorphism.”

Consider the group of integers “Z”, and

the integers mod 2. The operation we’ll use in both groups is

addition. The integers are an infinite group, while

the integers mod 2 is a finite group with only 2 elements.

So at first you might think these two groups are completely different from each other.

But consider this. The integers can be broken into two sets: the even and odd integers.

If you add one even integer with another, you get an even integer.

If you add an even and an odd, you get an odd.

If you add an odd and an even, you get an odd.

And finally, if you add two odds, you get an even integer.

Now compare this with the integers mod 2. In this group, 0 plus 0 equals 0.

0 + 1 equals 1. 1 + 0 equals 1.

And 1 + 1 equals 0. Do you notice a similarity?

If you replace “Even” with 0, and “Odd” with 1, then these say the exact same thing.

This an interesting observation. By splitting the integers into two sets - evens and odds

- we see these two sets behaved exactly the same as the group of integers mod 2. We can

express this observation in mathematical terms by defining a function “F” from the integers

to the integers mod two. The function is simple: if an integer is even, map it to zero. If

an integer is odd, send it to one.

Is it possible to create a reverse function that encodes our observation that evens and

odds behave like the group of integers mod 2?

The answer is … no. Suppose we tried to create a function G from

the integers mod 2 to the integers. We’ll send “0 mod 2” to some integer

“x”, and “1 mod 2” to some integer “y”.

Since “0 + 0 = 0 mod 2”, we want “x + x = x” in the integers.

This is because if we are comparing groups, then the behavior of elements in the output

should be similar to the behavior of the inputs. Solving the equation “x + x = x” gives

us “x = 0”. So G maps “0 mod 2” to 0.

Next, “1 + 1 = 0 mod 2”, so we want “y + y = 0” in the integers.

Again, this is because we want the inputs and outputs to behave similarly.

If we solve “y + y = 0” we get “y = 0” So G also maps “1 mod 2” to 0…

The function G is rather trivial. It sends everything in the “

integers mod 2” to 0… In this direction, we lose the group similarity between the evens

and odds and the “integers mod 2.” Keep this in mind when using a function to compare

two groups. The direction can matter.

Let’s see another example where we take two groups and compare them for similarities.

The integers mod 4, with addition as the group operation… and the group of four numbers

1.. -1.. i.. and -i with multiplication as the operation. To compare them, let’s look

at the Cayley table for both groups.

First, the group of integers mod 4 with the

operation of addition. The way you say this aloud is “the group

of integers mod 4 under addition.” This is the jargon used in math for specifying

the group operation. In the upper left corner, write the group

operation. Then in the first row and first column, list

all the elements in the group. Next, we go through and compute all possible

16 operations. 0 + 0 = 0 mod 4…

0 + 1 = 1 mod 4…

0 + 2 = 2 mod 4…

0 + 3 = 3 mod 4…

Continuing, 1 + 0 = 1…

1 + 1 = 2…

1 + 2 = 3…

and 1 + 3 = 0. And we’ll just go ahead and fill out the

remaining 8 squares…

Next, let’s make the Cayley table for the

numbers 1, -1, i and -i under multiplication.

To finish, we quickly perform the 16 multiplications.

We now have the Cayley tables for these two groups. They’re both the same size - they

each have 4 elements. But do they behave similarly? Or are they different and unrelated to each

other? We’ll answer this question visually by coloring the squares.

Each group has an identity element.

For the integers mod 4 it’s the number 0. And for the second group it’s the number

1. Let’s highlight the identity elements red.

To help you see the pattern, let’s switch “-1” and “i” in the group on the right.

The red squares now form a similar pattern in both Cayley tables.

Let’s move on to the next element in the group of integers mod 4, the number 1.

We’ll go through and color all squares containing a 1 green.

We do the same thing to the group on the right. The next uncolored element is “i”, so

let’s highlight all squares with an “i” green.

Notice the pattern of green in both tables is the same.

Let’s keep moving. In the integers mod 4, highlight all the 2’s

blue. And in the next group, highlight the “-1s”

blue. Once more, the blue pattern on the left is

the same as the blue pattern on the right. And we’ll fill in the squares with the last

number using purple. Same pattern, again.

By coloring the squares a different color for each element, we can clearly see the patterns

for both Cayley tables are exactly the same. They’re identical groups!! They just use

different elements and a different operation, but other than that, the groups are equivalent.

In both groups, if you combine a green element with a blue element, you get a purple element...

Similarly, a blue combined with a blue gives you a red, and so on.. Any such statement

about one table applies to the other. We say these two groups are “isomorphic”, which

means “equal form.”

Now that we’ve seen a couple of examples, let’s talk about things more generally.

Suppose we have two groups G and H. They can be ANY kinds of groups: finite, infinite,

commutative, non-commutative, you name it. Now pick any 2 elements X and Y in the group

G. If we combine X and Y, we get a third element

in G. We’ll pronounce this “X times Y” even

though the operation can be something completely different than multiplication.

For the groups G and H to have similar group behavior, X, Y, and “X times Y” in G must

correspond to elements in the group H.

The mathematical way to write down this correspondence

is with a function. So we want a function F from G to H that sends

this part of the multiplication table for G to a similar part of the multiplication

table for H. F sends X to “F of X”, “Y” to “F

of Y”, and “X times Y” to “F of X times Y”

Here’s the critical observation. In the table for H, this square must also

contain “F of X” times “F of Y” In other words, “F of X” times “F of

Y” must equal “F of X times Y”. This type of function is the mathematical

tool we use to compare two groups.

We now have a tool for comparing two groups G and H. It is any function F from a group

G to a group H such that “F of X times Y” equals “F of X” times “F of Y.” We

call F a “group homomorphism”, or just a “homomorphism” for short. One thing

to be careful about is the operation on the left is the group operation in G, while on

the right it’s the group operation in H.

As an example, define the function F from the integers Z to itself by “F of X equals

2X.” In this example, the group operation is regular

addition. To check if it’s a homomorphism, we only

have to show that “F of X + Y” equals “F of X” plus “F of Y”

From the definition of F, “F of X+Y” equals 2 times “X+Y”.

And on the right-hand-side, “F of X” equals “2X”, and “F of Y” equals “2Y”.

“2 times X+Y” DOES equal 2X plus 2Y, so F is a homomorphism.

Notice that the output of F is the set of even integers.

The even integers are a subgroup of the integers. We’ll explore this and other properties

of homomorphisms later.

After Groups, the idea of a Homomorphism is quite possibly the most important concept

in Abstract Algebra. It allows us to connect similar groups and identical groups. Homomorphisms

are also an essential tool for identifying the fundamental building blocks of groups.

And as we move forward through abstract algebra, this idea will recur again and again. We will

introduce a new object - rings, fields, vector spaces - and then define homomorphisms between

them.

The set of people who support us on Patreon is a finite but important group. They are

homomorphic to the Magic Group. This famous group is something I just made up as a way

to encourage you to support us financially on Patreon…{magic!}