Direct Products of Groups

There are two sorts of products on groups that I’d like to discuss. Today I’ll talk about direct products.

The direct product says that we can take two groups, form the Cartesian product of their sets, and put the structure of a group on that. Given groups $G$ and $H$ we form the group $G\times H$ as the set of pairs $(g,h)$ with $g$ in $G$ and $h$ in $H$ . We compose them term-by-term: $(g_1,h_1)(g_2,h_2)=(g_1g_2,h_1h_2)$ . It can be verified that this gives us a group.

There’s a very interesting property about this group. It comes equipped with two homomorphisms, $\pi_G$ and $\pi_H$ , the “projections” of $G\times H$ onto $G$ and $H$ , respectively. As one might expect, $\pi_G(g,h)=g$ , and similarly for $\pi_H$ . Even better, let’s consider any other group $X$ with homomorphisms $f_G:X\rightarrow G$ and $f_H:X\rightarrow H$ . There is a unique homomorphism $f_G\times f_H:X\rightarrow G\times H$ — defined by $f_G\times f_H(x)=(f_G(x),f_H(x))$ — so that $\pi_G(f_G\times f_H(x))=f_G(x)$ and $\pi_H(f_G\times f_H(x))=f_H(x)$ . Here’s the picture.

Universal Property of Products

The vertical arrow from $X$ to $G\times H$ is $f_G\times f_H$ , and I assert that that’s the only homomorphism from $X$ to $G\times H$ so that both paths from $X$ to $G$ are the same, as are both paths from $X$ to $H$ . When we draw a diagram like this with groups on the points and homomorphisms for arrows, we say that the diagram “commutes” if any two paths joining the same point give the same homomorphism between those two groups.

To restate it again, $G\times H$ has homomorphisms to $G$ and $H$ , and any other group $X$ with a pair of homomorphisms to $G$ and $H$ has a unique homomorphism from $X$ to $G\times H$ so that the above diagram commutes. This uniqueness means that has this property is unique up to isomorphism.

Let’s say two groups $P_1$ and $P_2$ have this product property. That is, each has given homomorphisms to $G$ and $H$ , and given any other group with a pair of homomorphisms there is a unique homomorphism to $P_1$ and one to $P_2$ that make the diagrams commute (with $P_1$ or $P_2$ in the place of $G\times H$ ). Then from the $P_1$ diagram with $P_2$ in place of $X$ we get a unique homomorphism $f_1:P_2\rightarrow P_1$ . On the other hand, from the $P_2$ diagram with $P_1$ in place of $X$ , we get a unique homomorphism $f_2:P_1\rightarrow P_2$ . Putting these two together we get homomorphisms $f_1f_2:P_2\rightarrow P_2$ and $f_2f_1:P_1\rightarrow P_1$ .

Now if we think of the diagram for $P_1$ with $P_1$ itself in place of $X$ , we see that there’s a unique homomorphism from $P_1$ to itself making the diagram commute. We just made one called $f_2f_1$ , but the identity homomorphism on $P_1$ also works, so they must be the same! Similarly, $f_1f_2$ must be the identity on $P_2$ , so $f_1$ and $f_2$ are inverses of each other, and $P_1$ and $P_2$ are isomorphic!

So let’s look back at this whole thing again. I take two groups $G$ and $H$ , and I want a new group $G\times H$ that has homomorphisms to $G$ and $H$ and so any other such group with two homomorphisms has a unique homomorphism to $G\times H$ . Any two groups satisfying this property are isomorphic, so if we can find any group satisfying this property we know that any other one will be essentially the same. The group structure we define on the Cartesian product of the sets $G$ and $H$ satisfies just such a property, so we call it the direct product of the two groups.

This method of defining things is called a “universal property”. The argument I gave to show that the product is essentially unique works for any such definition, so things defined to satisfy universal properties are unique (up to isomorphism) if they actually exist at all. This is a viewpoint on group theory that often gets left out of basic treatments of the subject, but one that I feel gets right to the heart of why the theory behaves the way it does. We’ll definitely be seeing more of it.

February 27, 2007 - Posted by John Armstrong | Algebra, Group theory, Structure of Groups, Universal Properties

13 Comments »

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I wonder if universal properties could be made easier for beginners to grasp if more of the structure of the concept was displayed directly in the language. For example, an attempted definition of direct product:

$H$ has the direct product property ‘flabbily’ for $F, G$ via functions $f, g$ iff $f$ is a homomorphism from $H$ to $F$ and $g$ is a homomorphsim from $H$ to $G$ .

$H$ is has the direct product property universally for $F, G$ via functions $f, g$ if it does so flabbily, and for any other $H', f', g'$ which also does so flabbily, there is a unique homomorphism $h:H'\rightarrow H$ such that $f'=fh$ and $g'=gh$ .

$H, f, g$ are a direct product of $F, G$ if $H$ has the direct product universally for $F, G$ via $f, g$

Grammatically, the first part (the ‘flabby’ version of the property) is supposed to be a predicate with four arguments, corresponding to various bits of the cone construction: the apex, which is involved in the unique arrow, the ingredients (objects and arrows) of the base, the property that characterizes how these ingredients and the `via’ arrows are supposed to be related, and the via-arrows that connect the apex to the ingredient objects.

Then a lot of ‘verb-phrase anaphora’ is used in defining the strict/universal part, in order to gammatically display the fact that the concepts from the flabby part are being re-used.

Maybe this is too cumbersome to be useful to beginners, but, speaking for myself, I was never able to understand the usual verbiage concerning universal properties until I sort of understood how cones/limits worked. So the idea here is to express the structure of the cone (and cocone) ideas more directly than usual in the language, without dragging in as much abstraction.

Comment by MathOutsider | October 9, 2007 | Reply
Well, MO, that’s an idea that actually shows up in some circles. Witness the term “weak Natural Numbers Object” in topos theory. Unfortunately, “weak” already tends to mean something else in other areas of category theory…

Comment by John Armstrong | October 9, 2007 | Reply
That’s why I chose ‘flabbily’ – doesn’t sound very nice but everything else I could think of that sounded better already had some other meanings that even I have encountered, although not necessarily understood.

Comment by MathOutsider | October 10, 2007 | Reply
Maybe smoother reformulation of 6, which is supposed to follow the standard formulation a bit more closely:

$C, f, g$ is a flabby product of $A, B$ iff $f:C\rightarrow A, g:C\rightarrow A$ .

$C, f, g$ is a (real/universal) product of $A, B$ if it is a flabby one such that for any (possibly other) flabby one $C', f', g'$ there is a unique $h:C'\rightarrow C$ s.t. $f'=fh, g'=gh$ .

‘wannabe’ would perhaps be an alternative to ‘flabby’. Well only the actual beginners can judge whether something like this is helpful, assuming it isn’t actively misleading.

Comment by MathOutsider | October 10, 2007 | Reply
The (co-)universal property behind commutator subgroups in the above tediously explicit format:

$A, f$ is a wannabe ‘abelianator’ of group $G$ if $A$ is an abelian group and the image of $G$ under $h$ .

$A, f$ is a co-universal abelianator of group $G$ if it is a wannabe, and, for any other such wannabe $A',f'$ , there is a unique group homorphism $h:A\rightarrow A'$ such that $f'=hf$ .

Putting it in this form caused me to see that the relevant property was co-universal rather than just universal, tho presumably people with more talent or experience would see this immediately.

Reformulating in terms of the subgroup embeddings rather than homomorphisms between quotients, things get rearranged a bit:

$C,\iota$ is a wannabe-commutator of $G$ if $\iota$ is an injection of $C$ into $G$ and $G/C$ is commutative.

$C, \iota$ is a (co?)universal commutator of $G$ if it is a wannabe, and for any other such wannabe $C',\iota'$ , there is a unique $h:C\rightarrow C'$ such that $\iota=\iota' h$ .

Here general format of a (co-)cone isn’t being followed, since the universal object’s arrows are factoring thru the wannabe’s rather than vice-versa. Don’t know what the terminology for this is.

Comment by MathOutsider | October 12, 2007 | Reply
Well, no. In that case it’s not really a cone. It’s a universal object in some category — one which you described correctly — but not a category of cones.

Limits and colimits are terminal and initial objects in categories of cones and cocones, but not all universal properties comes from these sorts of categories.

Comment by John Armstrong | October 12, 2007 | Reply
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