The First Five Symmetric Groups

Nathan Carter

This page shows Cayley diagrams for the first five symmetric groups. Image data was exported from a tweaked version of Group Explorer and then pushed into Asymptote for rendering.

I had difficulty getting a decent Cayley diagram for S5 for several years, and then thought of an idea that made it possible. This page shares the results, leading up to S5 using S1 through S4. (My apologies that Sn on this page does not subscript the n; blame the wiki, or my knowledge of it.)

S1

S1 is boring, because it is the trivial group. Here is its Cayley diagram:

Cayely diagram of S_1

S2

S2 is slightly less boring, but is still cyclic. Here is its Cayley diagram:

Cayely diagram of S_2

S3

Here's where things really get interesting. The smallest nonabelian group, S3 is a fabulous place to find all sorts of interesting phenomena. The one that concerns us right now is that, unlike smaller groups, its Cayley diagrams look different depending on which generators you choose to represent by the arcs/arrows.

The particularly helpful technique that this page uses (which makes possible the image of S5 below) is that I will use a minimum number of transpositions to generate each symmetric group. So rather than show S3 as I would if I were emphasizing that it's also D3, using an order-2 and an order-3 element, I will show it using two order-2 elements, corresponding to the permutations (1 2) and (2 3), in cycle notation.

Cayely diagram of S_3

S4

Following the same convention for S3, S4 can be generated by the permutations (1 2), (2 3), and (3 4). We now move up into three dimensions, and thus I provide not only a static image but an animation you can watch that will better show you the depth of the figure. (Notice the copies of S3 inside.)

Cayely diagram of S_4

To see an animation of this diagram (as an animated GIF, 1.34MB), click Animation of S4.

S5

And now the real purpose of this page. In order to see S5, I followed a rather convoluted process, that went like this.
1. I wrote a Python program to generate a Pajek file describing S5 as a network with 120 nodes and 180 arcs.
1. I opened the file in Pajek and asked it to perform a 3D Fruchterman Reingold Energy layout. The result was beautiful, but Pajek doesn't do multi-colored edges or saving of animations that I know of. So...
1. I saved the Pajek file and wrote another Python program to read it back in and generate some Asymptote code that could create a PDF/PNG/GIF/whatever. I then used Asmyptote's animation feature to make the animation. (All images on this page were made using Asymptote, most also with the help of data exported from Group Explorer.)
1. I then ran into a problem. When I tried to run the Asymptote animation generation, my computer kept running out of memory. On the Asymptote help forum I ran across a kindly soul who helped me out, making my code more efficient, and running it on his machine to generate the lovely animated GIF you can download below. Thanks, Philippe!
It resulted in the image below, and the associated animation.

Cayely diagram of S_5

To see an animation of this diagram (as an animated GIF), which I highly recommend because a static image is quite incomprehensible, click Animation of S5.


Related

Wiki: Gallery
Wiki: Group Files