# Lists of small groups

I am looking for lists of small

-solvable groups

-nilpotents groups

-simple groups.

By this I mean, do there exist lists of all the groups of order smaller than $n$, for $n$ reasonably big, satisfying one of the above properties?

Does anyone know a reference for this?

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Install GAP: it has a library of all small groups, and knows those properties. –  Mariano Suárez-Alvarez Apr 27 '11 at 17:02
You can get GAP from its webpage at gap-system.org –  Mariano Suárez-Alvarez Apr 27 '11 at 17:04
For nilpotent groups, you will quickly be disappointed. Every 2-group is nilpotent, and the number of those grows very fast. Conway's article "Counting groups: gnus, moas and other exotica" www.math.auckland.ac.nz/~obrien/research/gnu.pdf gives some numbers: for n=256, the number of groups of order n is already way over 9000, for 1024 it is 49487365422 and for 2048, the exact number is already unknown, but it is more than 1774274116992170. So your "reasonably big n" cannot be that big. For simple groups, I'm sure you'll enjoy madore.org/~david/math/simplegroups.html –  Maxime Bourrigan Apr 28 '11 at 1:35
I know I saw a book entitled "Groups of order $2^n$, $n\leq 6$" in a library; it is referenced in the Wikipedia article in the accepted answer. –  Allen Knutson Apr 28 '11 at 3:35

In http://en.wikipedia.org/wiki/List_of_finite_simple_groups there is a list of simple groups up to 9828. Moreover, the GAP small group library contains all groups of order less than 2000.

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Except those of order 1024! :) –  Mariano Suárez-Alvarez Apr 27 '11 at 17:08
Let's hope it doesn't turn out that these are the groups, Thomas is most interested in. –  Ralph Apr 27 '11 at 19:06
I remember being astounded at a talk when I first heard about exactly what percentage of small (simple) groups are 2 groups. It makes sense, but still shocking. –  Sean Tilson Apr 27 '11 at 23:54
@Sean You must be confusing something. The only simple 2-group is the cyclic group of order 2. The same goes with 2 replaced by any other prime. –  Alex B. Apr 28 '11 at 0:08
I agree GAP is the way to go if you want a list of groups of order less than n, and GAP makes it easy to filter on the properties you mentioned... but I'm kind of surprised no one mentioned the Atlases: brauer.maths.qmul.ac.uk/Atlas/v3 or en.wikipedia.org/wiki/ATLAS_of_Finite_Groups (I'm also surprised no one flagged this question as too elementary. Personally, I think this information is very useful, but I've seen less basic questions rejected on MO. Probably this should be community wiki, but I defer to the MO experts.) –  William DeMeo Apr 28 '11 at 7:33

This kind of question has a long history (pre-computer), but it's always been difficult to get much intuition about solvable groups just from looking at a big list. In the case of simple groups the old search by Marshall Hall and others for those of order less than a million was a more sensible fishing expedition: such groups are sparse but interesting, while ruling out certain orders could be suggestive. Computers (and GAP in particular) have improved numerical searches a lot, but there is only so much to be learned this way.

For a sense of older times, there is the ambitious book project:

MR0168631 (29 #5889), Hall, Marshall, Jr.; Senior, James K. ⋆The groups of order 2^n (n ≤ 6). The Macmillan Co., New York; Collier-Macmillan, Ltd., London 1964 225 pp.

For modern times, there is a 2002 survey with plenty of references:

MR1935567 (2003h:20042), Besche, Hans Ulrich (D-AACH-DM); Eick, Bettina (D-BRNS-G); O’Brien, E. A. (NZ-AUCK), A millennium project: constructing small groups. Internat. J. Algebra Comput. 12 (2002), no. 5, 623–644.

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Thanks for reminding of the Hall-Senior book. I don't know much about the content but the outer format of the book is quite impressing (if I remember correctly its width is at least half a meter). –  Ralph Apr 28 '11 at 1:01