Generators and Relations
Now it’s time for the reason why free groups are so amazingly useful. Let be any set, be the free group on , and be any other group. Now, every function from into extends to a unique homomorphism . Just write down any word in , send each letter into like the function tells you, and multiply together the result!
So what does this get us? Well, for one thing every group is (isomorphic to) a quotient of a free group. If nothing else, consider the free group on the set of itself. Then send each element to itself. This extends to a homomorphism from to whose image is clearly all of . Then the First Isomorphism Theorem tells us that is isomorphic to . That’s pretty inefficient, but it shows that we can write like that if we want to. How can we do better?
In a moment we’ll need this little technical construction. Remember that not every subgroup is suitable as the kernel of a homomorphism — it needs to be normal. We can beef up any subgroup to a normal one in a straightforward way, though. First notice that the intersection of any collection of normal subgroups is a normal subgroup again (check it). Now if is a group with subgroup , consider the collection of all normal subgroups of that contain . There’s always at least one, since itself is an example. Now take their intersection and call it . This is the smallest normal subgroup of containing , since it’s contained in every other one. We call it the “normal closure” of in .
Okay, so what we want is a free group and a normal subgroup of so that is isomorphic to . By the previous paragraph, we can settle for any subgroup and take its normal closure to get our . But this subgroup is a group in its own right, and is itself the image of some homomorphism from another free group .
Now we’re getting somewhere. Let and be any two sets, so we have the free groups and . Now take a function from to . This extends to a homomorphism, whose image is some subgroup of . Take the normal closure of this subgroup and get the quotient . We can get any group at all like this! We call the elements of “generators” and the words in the image of “relations”.
The best situation is when a group is “finitely presented” — and are finite sets. In this case we just have to write down the names of elements of , the words on in the image of , and the machinery above gives us back a group. Quite a lot of groups arise like this. We write such a presentation as for the generators and relations .
- The cyclic group is .
- The symmetric group is .
- The free group is .
So this should provide a way to tell if groups are isomorphic, right? Wrong. You might think that you should be able to tell when two presentations give isomorphic groups, but in fact it’s known that there’s no way to tell in general. The hangup is in what’s called the “word problem” for a group: given a presentation of a group and a word on the generators, do the relations make the word correspond to the trivial element of the group? It’s known that there is no method that solves this problem for all groups, and that many groups have no method for solving it at all.
Still, presentations by generators and relations are extremely useful for understanding the structure of a given group. As for free groups we can specify a homomorphism from by defining it on a generating set, though now we have to check that the relations are respected so the image of an element of doesn’t depend on the word we use to represent it. We can also prove facts about elements of “by induction”: show a statement holds for a generator and that composition and inversion preserve the truth of the statement. We can’t do everything we’d like with presentations, but they’re still one of the most concrete ways to actually get our hands on a group.