This has been nearly finished for a long time. I thought I should finally release it on Father’s day, in honor of my dad who has made several attempts to understand group theory despite an ocean and 8 time zones separating us.
I am always unsure of how to explain what I do to non-mathematicians. In order to understand my research someone first has to understand some group theory, in particular character theory of finite groups. Group theory is a part of abstract algebra that deals with symmetry. For that reason it shows up throughout mathematics as well as physics, chemistry, cryptography, etc. Today, I won’t explain group theory, but I’ll give you a pretty picture of it’s greatest achievement: The Classification of Finite Simple Groups.
A major goal of group theory has been to characterize what types of finite groups can exist. That is to say, what sorts of symmetries of finite objects can exist. The mathematical community has succeeded in part by classifying all the finite simple groups. Stated simply (pun intended), simple groups are groups which cannot be constructed from smaller pieces. It is not true that they are in fact simple in the conventional sense of the word. The problem of how to build groups from smaller pieces seems hopeless to understand in full generality. This is quite similar to the problem of understanding all molecules compared to understanding all elements.
There is a good article from a few years ago about group theory and the CFSG, as it is known colloquially. It is the longest proof ever written, weighing in at over 10,000 pages in it’s original form of about 500 journal articles. There is currently an initiative underway to rewrite the entire proof in a more concise and relaxed style. The first 6 volumes, and 2 volumes on quasi-thin groups, have already been published, in case someone wants to get me a Christmas present. :-)
The Periodic Table
Probably more familiar to most people is Mendeleev’s Periodic Table of the Elements. This is a listing of all the known elements, in order of increasing atomic number, into a table so that the elements in a column have similar properties (due to the configuration of outer electrons). Sometimes we forget just how amazing the periodic table is. After all, why should the chemical properties of the elements repeat periodically when ordered by atomic number?
The periodic table is certainly ubiquitous in popular culture, with periodic tables of many different subjects appearing on the internet. This is no doubt due to the fact that nearly everyone is familiar with it from High School chemistry. Below are a sampling of the diverse nature of the periodic tables available.
- Elephants (get it?),
- Fishing Lures,
- Harry Potter,
- Ice Cream,
- New Deal Programs,
- Rejected Elements
- (Unmitigated) Disasters,
- Video Game Controllers,
- Women in Sci/Fi,
- Xbox Games,
- and many many many more.
I just threw in for free Abecedarium of Periodic Tables. Now I just need a Periodic Table of Abecedaria… or a Periodic Table of Periodic Tables. Maybe an Abecedarium of Abecedaria should be next?
A periodic table strikes me as a good way to display the finite simple groups since, apart from 26 sporadic groups and the trivial group (which in not usually included), they all fall into families which can be arranged as the columns of the table. Moreover, there is one family that is completely different than the others (the cyclic groups of prime order), which corresponds to the noble gases. The alkali metals are also fairly different and have a parallel in the alternating groups which are quite different than the groups of Lie type.
Despite the abundance of periodic tables, I could never find a Periodic Table of Finite Simple Groups. So I decided that I had to take action, and I created The Periodic Table of Finite Simple Groups.
How I created it
My guiding principles in creating the Periodic Table of Finite Simple Groups were to be visually attractive, look as much like the real periodic table as possible, and be laid out in a logical manner. Initially I was inspired by Ivan Griffin’s (nice name BTW) periodic table written in LaTeX/TikZ, and early versions of my table were based on it. At this point, our versions bear no resemblance to each other apart from the fact that they are both written in LaTeX and TikZ and they use some of the same colors.
Since I wrote it in LaTeX with TikZ, the result is a pdf. The paper size is not standard because it was easier to construct that way. However zooming to fit on A4, or other A sized paper should work well. US Letter will be slightly more awkward, but should pose no real problems.
Perhaps the most difficult part, besides deciding on a layout, was getting the orders (that means the size of the group) to look good. I wanted them formatted with spaces separating groups of 3 digits, and broken across lines. It was also obvious that they would need different sized fonts. I made the formatting automatic based on the number of digits so that, for example, less than 15 digits goes on one line, less than 40 on two lines, and so on. While such a thing is certainly possible to write in pure (La)TeX, I thought it would be much easier to write in lua using LuaLaTeX. It turns out it wasn’t too difficult. By far the hardest part was making sure that things didn’t get expanded incorrectly, a task made more difficult by my desire to keep everything in a single file.
The Table Explained
In each column the groups increase in size going down, and as a very general rule I tried to put smaller groups to the left. More important however, was that similar families be next to each other. The same logic appears in the arrangement of the sporadic groups, where, for instance, all the Mathieu groups are together despite this causing orders to skip around a bit.
I also considered the non-classical groups to be “less important” and so there are fewer rows of them to better match the look of the real periodic table. They also have larger orders so it makes sense to include fewer of them on the table.
Since there are no “sporadic” elements, I had to decide what to do with the sporadic simple groups. These are groups that don’t fall into any of the other families. At one point I put them in the upper right corner with a jagged boundary like the non-metals. This made the table resemble the real periodic table, but it had more rows and less columns so it looked rather “thick”. Unfortunately, it didn’t make as much sense from an algebraic point of view, so I placed them where the lanthanides and actinides are found instead. This is slightly misleading, because if enough new elements were to be discovered there would be another row in that section. But you can’t have everything perfect. After all, there are an infinite number of simple groups, and only a finite number of elements.
I put the legend in the lower left like many periodic tables I have seen. I had a lot of space above the table, which I filled with Dynkin diagrams. Dynkin diagrams are the starting place for classifying the groups of Lie type. Note that the twisted groups of Lie type (those with a superscript to the left of the letter) arise from symmetries in the corresponding Dynkin diagrams.
The last row contains generic information and formulas for the family as a whole. Each cell contains some information, namely the order on the bottom, and the symbol most commonly used to denote it. The upper left will contain other symbols by which they may be known. For example the family of groups is also known as , and . The same is true for the sporadic groups, the monster is known as , , and . For the rest of the groups, the upper left contains “sporadic isomorphisms”, e.g. is isomorphic to . The table is large enough that all such sporadic isomorphisms are on the table, except of course for the infinite family of isomorphisms . Such groups are listed only once: in the left-most column to which they belong.
I have of course made every attempt to provide accurate information, but if you notice any mistakes or suggestions for improvements, please let me know: I did use wikipedia as one of my sources.