## 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