On this web page we present the data from the third run of the computer calculation of the mod-2 cohomology of groups of order 8, 16, 32 and 64. All of the calculations were made using the MAGMA computer algebra system. The groups are indexed by their Hall-Senior Numbers.
The first run of the cohomology rings was terminated in July of 1997. The cohomology rings of all but five of the groups of order 64 were successfully calculated. The failure to compute the last five were all due to problems with the Groebner basis machinery used to make minimal sets of relations among the generators of the cohomology. To view the results of the first run click here. The second attempt at the calculations was begun in February of 1998. As of early November 1998, all but six of the cohomology rings of the 267 groups of order 64 had been calculated. Some partial results had been posted for the cohomology rings of the remaining six groups. As in the first run the problems have been mostly in the Groebner basis computations. For the third run we wrote programs that circumvented these problems. To view the results of the second run click here. The user should be warned at this stage that the results of the computation on the third run will, in many cases, not look like the results from the first or second runs. There are several probabilistic elements in the calculation that cause different choices of variables for the cohomolgy rings in many of the calculations. The computed cohomology rings from the first, second, and third runs will certainly be isomorphic, but direct comparisons cannot be made.
Brief explainations of some of the terms and concepts that appear in the output of the calculations of the mod-p cohomology rings is in the text of the web page for the first run. Here is a description of the data that we present in these HTML pages. Note that the computations are usually in a red font .
A description of the HTML pages of abelian groups.A description of the HTML pages of nonabelian groups.
This web page is very experimental and is not likely to be refereed or reviewed by anyone except the users. Please feel free to send me any suggestions for improvement as well as any misprints or mistakes that you might notice.
Most of the calculations that are posted were performed on an SUN ULTRA 60 Elite 3D, (toui). More notes on the details of the calculations are given below. I want to thank the National Science Foundation and University of Georgia Research Foundation for providing me with both the equipment and the time to work on this project.
All of the programs are written in MAGMA code and run on the MAGMA platform. Thanks are due to John Cannon and Allan Steel of the MAGMA project for numerous instances of help with the tools to make the programs work and for their enthusiastic support. I am currently working on adaptations of the programs that will be suitable for inclusion in MAGMA in the near future.
One of the aims of the project has been to test some of the conjectures and investigate some of the many questions related to the structure of group cohomology. Accordingly we point out a few of the findings of the calculations. These are statements which hold for all of the cohomology rings that have been calculated. It is likely that there are many more interesting features that have not been observed in the limited time that we have had to examine the data.
[Dep] J. F. Carlson, Depth and transfer maps in the cohomology of groups, Math. Z., 218 (1995), 461-468.
[Test] J. F. Carlson, Calculations of cohomology: Tests for completion, (to appear)
[Prob] J. F. Carlson, Problems in the calculation of group cohomology, (to appear)
[HaSe] M. Hall and J. K. Senior, Groups of order 2n, n less than or equal to 6, Macmillan (1964), New York