Imagine, if you will, that we have two groups which we will creatively call “Group 1”
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!}