Monday, September 17, 2012

What is a quotient group?

Group theory is all about permutations. The elements of a group are the various operations whereby you can re-arrange a set of elements. I used the word "elements" twice in that sentence, and I did it on purpose. It's confusing because the word "elements" refers to two different things: you have the "elements" A,B,C and D which we are going to shuffle around, and you have the operations such as "flips" and "rotation" which are the "elements" of the permutation group on the four elements (letters).

There are obvious way of getting around this confusion, but mathematicians have a way of being just a little too clever about these things. They do something which seems brilliant but turns out to be nasty in the end. They figured out that it's unnecessary to talk about the "elements" A,B,C, and D. Everything you need to know about the structure and properties of a group can be worked out by constructing the algebra of flips and rotations (F's and R's). You don't need to think about what is being flipped or rotated. I wrote about this yesterday.

It's all very elegant but it leads to a very awkward situation when you come to deal with the topic of quotient groups. Quotient Groups are something that makes a lot of sense in terms of permutations of elements like A,B,C and D. Let's consider the set of all permutations on these four letters; as we remarked yesterday, there are 24 possible re-shufflings (including the trivial one which leaves the letters as they were). What gets interesting is when we look at the set of functions of A,B,C, and D defined as:

O1 = AB + CD
O2 = AC + BD
O3 = AD + BC

I talked about these three functions last spring when I wrote about solving the fourth degree equation. The interesting thing about them is what happens when you re-shuffle the four letters. We've already noted that there are 24 possible re-shufflings of the letters A,B,C, and D. But you ought to be able to convince yourself that no matter what you do, the result of the re-shufflings is going to simply return the three functions you started with. They might be re-arranged: you might flip O1 and O2, or you might rotate O1 => O2 => O3. Or the thetas might just stay where they were. The overall effect is that from the twenty-four possible permutations of the four elements A,B,C and D, you have generated the six possible permutations of the three elements O1, O2, and O3. 

That's what a quotient group is. You get a quotient group whenever there exists a set of functions which is preserved by the action of the group elements...that is, any permutation on the A's, B's and C's returns the same set of functions (the thetas in my example) that you started with. The funny thing is I don't think you'll find it explained this way anywhere. The way it's traditionally done is by defining something called a normal subgroup and its cosets. You investigate the group action on these sets, and you identify the resulting structure as a "quotient group". You can look up quotient groups and normal subgroups on Wikipedia and you'll see that this is how it's done.

Just how different are these two approaches? I think they're very, very different. For one thing, the "normal subgroup" defined in the traditional approach has four elements, while my functional approach operates on three elements. It's different. Different approaches usually give useful insights, but I can't find my method being used anywhere. 


What disinguishes the two approaches is that I treat a group as a set of operations working on a set of target elements, whereas the sophisticated mathematical approach is to ignore the target elements and define everything in terms of the properties of the operations. The formal development of quotient groups uses this lean, efficient methodology. But at what cost?

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