More groups

The title reminds me of the occasion when my academic grandfather lectured to the London Mathematical Society on “Moore graphs”. Unfortunately, someone got it wrong, and his lecture was billed as “More graphs”.

But here I really do mean what I wrote.

The function gnu(n) (for Group NUmber) is the number of groups of order n (up to isomorphism). It is an extremely irregular function. It is known up to 2047, and of all the groups of these orders, more than 99% have order 1024. The number is about 5×1010, which is itself dwarfed by the number of groups of the next order 2048, which is about 1.77×1015. However, it is an important function, and precise values are very much needed, as well as providing a computational challenge to existing algorithms and computing facilities. See this article for comments and elaborations.

Now Olexandr Konovalov has set up a crowdsourcing project, the “Gnu Project“, to calculate further values of the function, filling in some gaps in the presently known values. He has made available a GAP package to enable anyone to contribute. (The GAP website is here, in case you need to download this free computational system for algebra and discrete mathematics.) According to Olexandr, a forthcoming release of GAP will be optimized so as to steamroll over problems of this type.

GAP road roller

Instructions for getting and using the package and contributing to the database are provided. Take a look, and lend a hand!

About Peter Cameron

I count all the things that need to be counted.
This entry was posted in doing mathematics, software and tagged , , . Bookmark the permalink.

2 Responses to More groups

  1. alexk says:

    I have just posted an answer to my earlier question “Most wanted reproducible results in computational algebra” which I have asked on Mathematics Q&A site from the Stackexchange framework: I believe enumeration of finite groups is definitely one of the problems which should be more reproducible.

  2. alexk says:

    As of today, we have 999 “holes” in the table of values of gnu(n) for n less or equal than 50000. This excludes several more submissions for which replication is now in progress. The next goal is to get less than 50 holes below 10000.

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