Groups Lecture 6

in which we meet the symmetric groups and their cycle notation.

Today we started on one of the most important examples of groups, the symmetric group. We proved that permutations on a set X (that is, bijections X\to X) form a group under multiplication, which we call \mathrm{Sym}X.  When X is finite, we might as well take X=\{1,2,\ldots, n\}, and then we write \mathrm{Sym}X=S_n, the symmetric group of degree n. We saw two different notations to write down elements of S_n: the “two row notation”, which gives an intuition when thinking about it as strings, and which is good for proving that |S_n|=n!; and the cycle notation, which is the most commonly used, because we can read off quite a lot of properties from it. We saw how S_3 can be viewed as a subgroup of any S_n for n>3, by just considering the numbers above three to stay fixed, and also how for example D_8 can be viewed as a subgroup of S_4. We started on the path to disjoint cycle notation, by proving that disjoint cycles commute. More on that next time.

Understanding today’s lecture

The symmetric group is one of the most important examples, so make sure you get used to it! Play around with both notations, perhaps translate one into the other, practice composing elements in both. You can easily make up your own elements. Also make sure you’re happy with how to view D_{2n} as a subgroup of S_n. We only had n=4 as an example, but you will easily see how it generalises.

Preparing for Lecture 7

Can you read off the order of a k-cycle? What about the order of the product of two disjoint cycles? And non-disjoint ones? Next time we are going prove that “disjoint cycle notation works”, meaning that every permutation can be written (essentially uniquely) as the product of disjoint cycles. Do you have any of your own ideas of how we might do this?

Going a little deeper

Here are some thoughts about cycle lengths: if you take a 4-cycle, say (1234), and square it, you get (1234)^2=(13)(24). Can you formulate that into a general rule? Say something like “the square of a 2n-cycle is …”? Or even “the lth power of a ln-cycle is ….”? Here the “is…” should be something of the kind “a k-cycle” or “the product of two k-cycles”, or whatever you think it turns out to be.

We said that X can also be an infinite set. Can you think of some useful notations for say \mathrm{Sym}\mathbb{N}? We will meet some more examples later, for example when X=\mathbb{C}_\infty, the compactification of the complex plane, or the Riemann sphere. Then we will get Möbius maps. Keep your eyes open throughout your learning of maths to find more examples.

 

 

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