$p$ -Group Cohomology Package¶

This is the documentation for our Sage package on the computation of modular cohomology rings of groups of prime power order.

Summary¶

The source code consists of Cython extension modules and Singular functions written by Simon King, C-programs and Gap functions written by David Green, and a modified version of parts of the Aachen C MeatAxe.

The package comprises a data base of the cohomology rings of all groups of order 64, and can access a repository of the cohomology rings of all groups of order 128, all but 6 groups of order 243, of the Sylow 2-subgroup of the Higman-Sims group, and of the Sylow 2-subgroup of the third Conway group. These data were produced with our package.

The standard way of creating cohomology rings is documented in pGroupCohomology. More details on the available methods can be found in the cohomology module. There are also five other modules used in the background, which may be less interesting to a casual user.

We follow Jon Carlson’s minimal resolution approach. For the key steps we use algorithms by Dave Benson and David Green.

Installation¶

Before installing our package, make sure that the Small Groups library of Hans Ulrich Besche, Bettina Eick and Eamonn O’Brien is installed in your copy of Sage. You can install it during a Sage session by

sage: install_package('database_gap')

Then, for installing the latest “official” version of our package (1.1), do

sage: install_package('p_group_cohomology')

The latest experimental version (1.2) can be installed on the command line by

sage -i http://sage.math.washington.edu/home/SimonKing/Cohomology/p_group_cohomology-1.2.spkg

We are sorry for the long installation time.

Acknowledgements¶

The development of the initial version of this SPKG was funded by by the German Science Foundation, DFG project GR 1585/4–1, and mainly accomplished at the Friedrich Schiller University Jena.

Since version 1.0.1, the further work on this SPKG is funded by Marie Curie grant MTKD-CT-2006-042685 and is pursued at the National University of Ireland, Galway.

We thank William Stein for giving us access to various computers on which we could build test the SPKG and on which some huge computations could be completed, and acknowledge the support by National Science Foundation Grant No. DMS-0821725.

We are also grateful to William Stein and David Joyner for critical comments and for testing the installation of our package on a large variety of platforms. Suggestions of Mikael Vejdemo Johansson and John Palmieri were very valuable for verifying the code on the computation of Massey products.

Versions¶

v1.0.1: First public version in GPL 2 or later (July 2009)

v1.0.2: Fixes some bugs (July 2009).

v1.1: Added restricted Massey powers and general Massey products (August 2009).

v1.2: Modified printing for cocycles; minor bug fixes and code improvements. New: Persistent Group Cohomology (bar codes), based on ideas of Graham Ellis and Simon King (October 2009).