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.
Group Theory
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 its 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 its 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.
- Atheists,
- Beer,
- Cleaning,
- Dictators,
- Elephants (get it?),
- Fishing Lures,
- Guitarists,
- Harry Potter,
- Ice Cream,
- Jazz,
- Knitting,
- Literature,
- Meat,
- New Deal Programs,
- Oregon,
- Pirates,
- Queers,
- Rejected Elements
- Swearing,
- Typefaces,
- (Unmitigated) Disasters,
- Video Game Controllers,
- Women in Sci/Fi,
- Xbox Games,
- Yo-Yos,
- Zombies,
- 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.
EDIT: After a few people have asked in the comments, I feel like I should just state explicitly that anyone is free to use it for non-commercial use. The only favor I ask is that you let me see the finished product if possible.
UPDATE: I have an eight year retrospective and now a metal poster is available.
UPDATE: Some books with the PToFSG.
Fantastic!
I’m glad that i am smart enough to realize how really brilliant you are. What a great Fathers day present.
Reblogged this on Guzman's Mathematics Weblog.
[…] En este enlace el autor nos explica algo sobre su creación y en este otro enlace la tenéis en pdf para descargar. […]
Your simple group periodic table is wonderful Could I please have permission to include it in slides for a maths lecture?
Yes, of course you may use it. Incidentally, I met your student Seyed Hassan Alavi at a conference (Finite Groups and their Automorphisms) a few years ago.
Many thanks for this. Hassan finished his PhD a couple of years ago – I saw him at a conference in Istanbul early this year.
you should study chemistry more carefully, periodic table of elements is based upon “Nuclear charge”
You’re right. I should have written more carefully. It’s ordered based on atomic number, not atomic weight. Thanks.
you’r welcome! But so far I’ve been trying to understand the elements of table :)
Is there somewhere that I can print this out at poster size, or get the tex file for it? I really want to put this on my office door or wall nearby!
I suppose that I can zoom it indefinitely because it’s all done in Tikz, so I retract my last comment/question. May I have permission to post this on my office door please?
You can definitely put in on your office door! All I ask is that you send me a picture when it’s done. :-) Seriously though, I have thought about printing it nice and big for my office (well cubicle now), but I haven’t done it.
I can send you the source file if you want to try tweaking it for larger sizes (though I think it will blow up pretty well).
I will give it a shot as is. And I will post a photo once I’m done! Thanks!
That’s awesome ! I ‘m from Brazil and I am a Bachelor student of mathematics yet. I love Group Theory and its periodic table of the finite simple groups is fascinating !
My group theory might be a bit rusty, but what part about this table is periodic? It looks to me like several groups/families arranged as columns, but in what way does 2A2(9) follow 2D4(2^2)…and is that the same way that C3 follows A6? Or rather, in what way does the sequence:
D4(2) -> 2D4(2^2) -> 2A2(9) -> C3 -> A6 -> A1(8) -> B2(4)
make sense?
If it doesn’t…then what is the “periodic” nature of this table?
You’re right, it’s not really periodic. It’s just an homage to the periodic table of elements. Each column is related though, which is the same as the periodic table of the elements—in that sense it’s probably better than many “period table of X” you’ll find. :-)
Twice I spotted “it’s” instead of “its” in “Group Theory” section: “it’s greatest achievement” and “10000 pages in it’s original form”.
In case you wonder why it should matter, especially to a mathematician, it’s explained in the rest of my comment (so if you agree it matters, you can stop reading here).
I assume you’re familiar with the definition of gamma function, converging only where real component is above certain value, the rest being defined by expanding it to the left, using the formula gamma(z+1) = z× gamma(z).
Now imagine that when talking about it, one would say that using the the above formula “the function can be expanded to the whole of imaginary plane except non-positive integers”. Everyone will understand the statement, but strictly speaking it’s wrong, it should say “complex plane” as the term “imaginary” only covers the numbers located on the y axis. Just like “it’s” versus “its”, it’s a “minor” difference that in certain contexts won’t affect understanding even though it’s wrong / doesn’t make sense.
Thank you. I admit that I have a problem with “it’s” and “its”. For some reason I just always type “it’s”. I have now fixed them.
The table is fantastic! I was wondering if you might give me permission to use it on a web page advertising an 2020 semester programme at the Isaac Newton Institute in the UK? It would appear on the Newton Institute’s website.
Yes, indeed you may. I really should put it up somewhere with a license I guess.
Thank you so much!
The webpage is now up here: http://www.newton.ac.uk/event/gra.
Unfortunately, they have a standard format for the webpage which makes the heading at the top of the image look like the name of our programme. Would it be ok if we cropped off the heading “The periodic table of finite simple groups” and just used the main table?
Of course! Thanks for the link. That’s really cool.
I would like to use the periodic table for finite simple groups in a write up for the NSF for our representation theory program at MSRI this past semester. The table is really great. Thanks for doing this.
Of course you can. Will it be online afterwards? I just like seeing how it turns out.
Incidentally, I read several of your papers when working on my dissertation.
The table is fantastic. May I use it for a review article about our Collaborative Research Center in the newspaper of the University of Kaiserslautern, Germany? Among other things, we do research on representation theory of finite groups. The table would help to make these things understandable to the public.
Of course. I would be interested in seeing how the final article turns out if you remember to send a link.
Your simple group periodic table is fantastic! Could I please have permission to include it in slides and a guide for a maths lecture?
Yes of course you can. I’m just a little curious, what is the lecture about?
The book is a small help guide for students of the Abstract Algebra subject that is taught at the Polytechnic University of Madrid and will be published by the University itself. If you give me your address, I can send you a copy. Thank you so much!
Wow, fantastic!
I am not a mathematician but I like group theory and this table looks really cool! Thanks for sharing it. May I print it on a shirt for my personal use? Thanks!
Yes, please do! And if you feel like it, I wouldn’t mind seeing a picture of you wearing it.
Hi Ivan. On a similar note, my 19-year-old son is a big fan of the table, and I’m wondering if you would permit me to print it on a mug for his birthday. Thank you for your inspiring work and for considering it!
Please do. That sounds really cool.
Sorry for bother you at here now. Can you show me how your shirt looks?
Sorry for reply to you again. Currently, WordPress won’t let you know email address of repliers. You can just host your picture somewhere online, put the link in comment/reply, just like Ivan told me.
[…] year, almost in time for Father’s day, I got a metal copy of My Periodic Table of Finite Simple Groups. I was contacted recently by Carl Edman who wanted to make one for himself, and asked if I would […]
Dear Ivan, Cheryl Praeger advised that we may use your periodic table as the background image on our website for her talk.
Here is where it’s displayed: https://www.matrix-inst.org.au/events-01/online-seminars/
Please let us know if there’s any issue.
That looks great! Thanks for letting me know.
Hello! Your table has inspired my son, who is a math student at university. Would you be willing to grant permission to print it on a mug as a gift for him? Thank you for considering it!
Yes! That sounds wonderful. Sorry I missed your comment, I don’t know how that could have happened. Hopefully, it’s not too late.
Hi Ivan, this is super cool! I am creating a presentation on my work on essential p-dimension; may I use your periodic table when I am talking about the classification of finite simple groups?
Yes, of course. I realize this comes probably too late—I hope you went ahead and used it.
Do you have a link to your presentation?
Now we long for a crossover metal plate between your detailed periodic table and this wikipedia stylized version :) : https://upload.wikimedia.org/wikipedia/commons/d/d7/Classification_of_the_finite_simple_groups.png
Hello! Sorry for posting reply in 2023. I’m very interested in math and found your table looks cool, so I printed the table on a shirt some days ago for personal wearing (mine is long-sleeved, may not same as Chao Feng’s one). IDK how to send a pic to you, but still thank you for your work!
Awesome! I’m glad you like it. I think it would be cool if you posted a comment pointing to it (I think it would have to be hosted somewhere else). Then everyone could see it. But it’s up to you of course.
Picture link is here: https://raw.githubusercontent.com/claybiockiller/Clayblock-random-pics/main/IMG_20230115_005048.jpg