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I was wondering whether or not there was an online encyclopedia of groups--finite or infinite. If there isn't, would you suppose that such a thing would be useful?

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Have you looked at hobbes.la.asu.edu/groups/groups.html? –  Gerry Myerson Apr 11 '12 at 4:24
Groups of order at most 12 are given at web.science.mq.edu.au/~chris/groups/appendix.pdf –  Gerry Myerson Apr 11 '12 at 4:27
The Atlas of Finite Group Representations looks relevant. –  anon Apr 11 '12 at 4:29
Nobody yet has mentioned Groupprops, the group properties wiki –  MJD Apr 11 '12 at 5:24
To expand on hardmath's comment, GAP (and anything that interfaces to GAP e.g. sage) has a SmallGroups library containing every group of order <=2000 except those of order 1024 --- this is over 400 million groups. It also has all groups of squarefree order, small cubefree order groups, all $p$-groups of order $\leq p^6$,... See gap-system.org/Manuals/doc/htm/ref/CHAP048.htm#SECT007 It also has databases of primitive permutation groups, classical groups, finite perfect groups, and more: gap-system.org/Manuals/doc/htm/ref/CHAP048.htm –  mt_ Apr 11 '12 at 6:47

4 Answers 4

up vote 5 down vote accepted

You may also want to read a nice article of Conway, Dietrich and O'Brien http://www.math.auckland.ac.nz/~obrien/research/gnu.pdf

And also the paper of Besche, Eick and O'Brien http://www.math.auckland.ac.nz/~obrien/research/2000.pdf which contains a table of the number of groups of order $n < 2001$.

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These papers look very interesting. I'll read them when I get the time. –  Chris Dugale Apr 13 '12 at 0:11

Groups of order less than 30 are at http://opensourcemath.org/gap/small_groups.html

Also, http://world.std.com/~jmccarro/math/SmallGroups/SmallGroups.html goes up to order 32.

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Another older reference is Marshall Hall, Jr, and James K. Senior, The groups of order $2^n\ \ (n < 6)$ (Macmillan, New York, 1964).

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Depending on your needs, Wolphram|Alpha or Mathematica itself might be helpful. See here for the computational overview, and here for the data function (open up the "More Information" for the sorts of things you can search.)

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