Tuesday, June 21, 2005

Math Post the Seventh: Kernels and Quotients

Last time I was talking about homomorphisms, the "nice maps" in the land of groups, and I gave some of the usual map definitions (injective, surjective, and so on). One of the things we (well, mathematicians -- I can't really answer for the rest of you) always want to know about maps is what the stuff over a particular point looks like; if you had a map to the integers, what stuff goes to three? Since this concept is so common, the set of stuff going to a particular point has a name: the fiber over that point. In the case of homomorphisms, the answer to this question has a simplifying wrinkle: the answer is the same for all points. To symbolify things a bit, let f:G->H be a homomorphism of groups and let h be in H. Now, say f(g)=f(g')=h, so both g and g' are in the fiber over h. Now, since f is a homomorphism, f(g^(-1)g)=h^(-1)h=e, so the difference between any two elements of the fiber over h are in the fiber over the identity; thus if you pick a particular element g in the fiber over h, and we call the fiber over the identity K (for reasons soon to become clear), the fiber over h is exactly gK. So all the information about the fibers is contained in the set K (called the kernel of the map f).

Now, we can say some quick things about K. First, K is a subgroup of G; two elements g and g' in K both map to the identity under f, so their product maps to the product of the identity with itself, which is still the identity, as is the inverse of the identity. However, K has one more special property: the left and right cosets of K are the same, that is for any g, gK=Kg (since both of these are precisely the fiber over the point f(g) in H). This implies the interesting property that gKg^(-1)=K, since this is the fiber over the identity again; this funny equation is called "conjugation by g". Now, if your group isn't abelian, conjugation may not fix all the elements of the subgroup K, but since K is a kernel, conjugation can only move an element of K to another element of K. Subgroups with the property of being fixed under conjugation are called "normal," in one of math's most badly overloaded terms.

Normal subgroups have one extra bonus fun property: their cosets form a group. Let N be a normal subgroup of G, and consider the cosets gN and g'N; if we try to multiply them in the straightforward way, we get gNg'N=gg'NN=gg'N (since N is closed under multiplication, NN=N). If N weren't normal, the left and right cosets wouldn't conicide, but since it is they do, and the cosets inherit all the group properties from the same properties of G. This new group is written G/N, and is called the quotient group of G by N. The easiest examples of this can be realized by the clock arithmetic groups Z/n (and now you understand the notation; because the integers are so familiar we omit the extra Z on the end) for various integers n; remember the cosets of the subgroup H=nZ are H, 1+H, 2+H, and so on up to (n-1)+H, and adding cosets is just adding the numbers out front and subtracting off n's if the number gets bigger than n. Of course, since Z is abelian, all subgroups are normal (the inverse just commutes through), so you need to be more careful in general.

G/N admits a nice map from G (an element g goes to the coset gN), nice in the way called "canonical," meaning that no choice is involved in determining the map -- we don't have to pick anything to make it up, it just comes to us. If you made a version of this map up, yours would be the same as mine. There are lots of non-canonical maps, and we'll meet plenty next time, but it's important to know about the differences in the kinds of nice maps.*

Next time: isomorphism theorems and factoring.

*And it's important to be specific about words; the inventors of a branch of mathematics called category theory, which is in some sense a language in which all other mathematics should be done, gave a definition for a kind of niceness they called "naturality." Fortunately for them (and us), the natural transformations of objects called functors are important and cool; unfortunately for us as students of math, people are sadly nonspecific with the word natural, rarely specifying when they mean the technical sense. One of my more trying math experiences was realizing after finishing a problem set (from the canonical algebraic geometry textbook) that when the word natural appeared in every problem it didn't just mean pleasant or straightforward but that there was suddenly much more work to do.

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