A recurring theme in science and mathematics is to find the fundamental building blocks
of something, then study the different ways you can put them together.
For example, in chemistry, the building blocks of everyday matter are atoms.
And there are different types of bonds,
which describe how they can be stuck together to make molecules...
In number theory, you learn that the building blocks of integers are prime numbers,
and that every positive whole number has a unique factorization into primes.
In group theory, the fundamental building blocks of finite groups are called simple groups,
and there are many ways you can piece them together to make larger, more complex groups.
One way is the direct product...
Suppose you have any two groups, G-1 and G-2.
These two groups can be finite or infinite, abelian or non-abelian.
The direct product is a way to combine these two groups into a new, larger group...
It’s the set of all pairs where the first component is from G-1,
and the second component is from G-2.
The notation for the direct product is the same as the Cartesian product of two sets.
The group operation is performed component-wise.
Let (a, b) and (x, y) be two elements in the direct product “G-1 times G-2”.
Then their product is defined to be the pair (a*x, b*y).
You just multiply the first components, then multiply the second components.
The identity element in this group is (E-1, E-2), where E-1 is the identity element
in G-1, and E-2 is the identity element in G-2.
Let G-1 be the group of integers under addition, and G-2 be the group {1, -1, i, -i} under multiplication.
The direct product of G-1 and G-2 is the set of all pairs
where the first component is an integer, and the second component is one of the four numbers in G-2.
Let’s pick two random elements from the direct product: how about (7, -1) and (-3, i).
If you wanted to, you could use a new symbol for the direct product group operation,
like an asterisk, but we’ll keep things simple and use multiplicative notation for the direct product.
To combine these two pairs, we ADD the first components, and MULTIPLY the second components.
This is because the group operation for G-1 is addition, and the operation for G-2 is multiplication
Doing this gives us (4, -i).
The identity element in this group is (0, 1).
For example, if you combine (5, -i) and (0, 1) you add the first components to get 5,
and you multiply the second components to get -i.
So the product of (5, -i) and (0, 1) is (5, -i).
We can check that this works for any element.
If (x, y) is an element in the direct product, then (x, y) times (0, 1) is (x + 0, y * 1) which equals (x, y).
So (0, 1) is an identity element.
It’s good practice to verify that the other group requirements are met.
For example, what’s the inverse of (12, -i) in this group?
Does every element have an inverse?
And does the associative property hold?
You can take the direct product of more than two groups.
For example, the direct product of G-1, G-2 and G-3 is the set of all TRIPLES
where the 1st component is from G-1, the 2nd component is from G-2
and the 3rd component is from G-3.
Like before, the group operation is performed component-wise.
The product of (a, b, c) and (x, y, z) is (a*x, b*y, c*z).
The identity element is (E-1, E-2, E-3) where E-1 is the identity element in G-1,
E-2 is the identity in G-2, and E-3 is the identity in G-3.
And there’s no reason to stop with three groups.
You can take the direct product of four.. five.. or ANY number of groups.
You can even take the direct product of an INFINITE number of groups!
As another example, let’s take the direct product of the integers mod 3 under addition…
the integers mod 5 under addition... and the integers mod 6 under addition...
The elements in this direct product are triples, with the first component from G-1,
the second component from G-2, and the third component from G-3.
The first group has 3 elements, the second group has 5 elements,
and the third group has 6 elements,
so the order of the direct product is 3 * 5 * 6 which is 90.
Since all three groups use additive notation, we’ll use additive notation for the direct product, too.
The sum of (1, 3, 4) and (2, 1, 3) is (0, 4, 1).
This is because 1 + 2 is 0, mod 3…
3 + 1 is 4 mod 5…
And 4 + 3 is 1 mod 6…
The identity element in this group is (0, 0, 0).
Here’s a question for you: What’s the inverse of (1, 3, 2) in this group?
We’ve now seen two examples of direct products: the integers with a finite group,
and the direct product of three finite groups.
It’s worth pointing out that if you take the direct product of a bunch of groups,
then if ANY one of the groups is infinite, the direct product will be infinite, too.
But what if all the groups are finite?
In this case, it depends.
If you take the direct product of a FINITE number of finite groups,
then you end up with another finite group.
In this case, the order of the direct product is the product of the sizes of the individual groups.
However, if you take the direct product of an INFINITE number of finite groups,
then you’ll end up with an infinite group.
Up to now, all the groups we’ve considered have been abelian.
Let’s now look at a direct product where one of the groups is NON-abelian.
For this example, we’ll look at the direct product of the real numbers under addition
and the symmetric group S-3.
While the real numbers are an abelian group, S-3 is a non-abelian group.
For example, let’s use cycle notation and look at the permutations A = (2 3) and B = (1 2).
A times B equals (1 3 2), while B times A equals (1 2 3).
So A times B does NOT equal B times A. This carries over to the direct product.
For example, if we combine (1, A) and (2.5, B), you get (3.5, (1 3 2)).
But if you reverse it and compute the product of (2.5, B) and (1, A) you get (3.5, (1 2 3)).
So the direct product is NOT an abelian group.
This example highlights another important point.
If you take the direct product of a several groups, then if any one of the groups is not abelian,
then the direct product is non abelian, too. Here's why:
Suppose we have a direct product, and we arrange the groups so that G-1 is a non-abelian group.
This means there are elements A, B in G-1 where A times B does not equal B times A.
Let’s now use this to show the direct product is non abelian.
Look at the elements X equals (A, E-2, E-3, etc.) and Y equals (B, E-2, E-3, etc.).
So the first components are A and B,
while the other components are the identity elements in their corresponding groups.
X times Y is (A*B, E-2, E-3, etc.) … and Y times X equals (B*A, E-2, E-3, etc.)
And since A*B does not equal B*A, these two values are different.
So the direct product is NOT commutative.
If ALL the groups are abelian, however, then the direct product will also be abelian.
You’ll use this fact when proving the Fundamental Theorem of Finitely Generated Abelian Groups.
In short, this says that ANY abelian group that is finitely generated
is the direct product of the integers or the integers mod N for various N.
The direct product is a useful tool for breaking some groups into simpler groups.
Notice I said “some groups” and not “all groups”?
This is because there are many groups which are NOT direct products of simpler groups.
It would be great if groups were that simple, but they’re not.
Luckily there are other tools for deconstructing groups.
And we’ll cover those in separate videos.
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