# Mod-*p* Group Cohomology Package¶

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

## Summary¶

The source code consists of

- Python and Cython extension modules as well as Singular and GAP functions written by Simon King.
- C-programs and Gap functions created by David Green and modified by Simon King.

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.

Since version 2.0, it can also compute the modular cohomology rings of non prime power groups. In particular, it allows for the computation of the modular cohomology for various primes of the first three Janko groups, of Mathieu groups 11, 12, 22 and 23, of the McLaughlin group, of SuzukiGroup(8), of the Higman-Sims group and of the third Conway group. Here are the computational results. It is planned (but not done yet) to include these cohomology rings in our repository.

The standard way of creating cohomology rings is documented
in `pGroupCohomology`

. More details on the available methods can be
found in the `factory`

,
`cohomology`

and
`modular_cohomology`

modules. There are
also five other modules used in the background, which may be less
interesting to a casual user.

The computation of the modular cohomology rings of non prime power groups is reduced to the case of prime power groups, possibly in several steps, by virtue of the stable element method. The cohomology computation for prime power groups is based on the construction of a minimal free resolution.

In both cases, we follow Jon Carlson’s basic approach to compute an approximation of the cohomology ring in increasing degree, and to use criteria to prove that at some point the approximation is actually isomorphic to the cohomology ring.

We use completeness criteria proposed by Dave Benson, David Green, Simon King and Peter Symonds. The construction of minimal free resolutions is based on an algorithm of David Green.

## Installation¶

The Small Groups library of Hans Ulrich Besche, Bettina Eick and Eamonn O’Brien is used at run-time. The default way of installing it into SageMath’s version of GAP is by

```
sage -i database_gap
```

By version 3.0, MeatAxe has been removed from the cohomology package. Instead, it is linked against SharedMeatAxe. So, it is a build-time dependency. You can install it in your copy of Sage by

```
sage -i meataxe
```

By trac ticket 18514, the cohomology package can then be installed in your copy of Sage by

```
sage -i p_group_cohomology
```

## Testing¶

The Cython and Python parts of the package have 100% doctest coverage, but
be warned that running the test suite requires a considerable amount of
time (easily one hour if a single thread is used). If the environment variable
`SAGE_CHECK`

is set to `yes`

, the test script is launched right after
installing the package. The same effect can be achieved by

```
sage -i -c p_group_cohomology
```

## Documentation¶

If the environment variable `SAGE_SPKG_INSTALL_DOCS`

is set to `yes`

, then
the documentation of our spkg is automatically created and put into
`SAGE_ROOT/local/share/doc/p_group_cohomology/`

.

## 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 Friedrich Schiller University Jena.

Since version 1.0.1, the further work on this SPKG was funded by Marie Curie grant MTKD-CT-2006-042685 and was pursued at the National University of Ireland, Galway. Since version 2.1.2, the project has returned to Jena and was funded by DFG project KI 861/2–1.

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.

We thank Mathieu Dutour Sikirić for his explanations how to keep track of large lists of double cosets in GAP. We are also grateful to the GAP support group for solving various technical problems that became imminent when dealing with non prime power groups.

We thank Peter Symonds for interesting discussions, in particular for suggesting to use the Poincaré series in a completeness criterion.

## Versions¶

- v3.0: Turn the cohomology package into a “new style spkg”. It is split into several smaller parts that are either using an autotoolized build system or are pip installable.
- v2.1.5, v2.1.6: Cope with several backwards incompatible changes in SageMath. Improved computation of the nil-radical, including a degree-wise computation. Methods is_nilpotent and nilpotency_degree for cohomology ring elements. Various improvements for the computation of depth and filter degree type.
- v2.1.4: Consequently compute parameters for the
completecohomology ring rather than only for the ring approximation. Better portability by using os.path. Replace SAGE_DATA by SAGE_SHARE. Use urllib2. Protocol output tells which instance’s method is being called.- v2.1.3: Improved heuristics to choose between Hilbert-Poincaré and Symonds criteria. Cope with Cython’s new “name mangling” for double underscore attributes. Allow storing of “permanent results” that are indexed in terms of data in Gap. In some situations, use a lower bound for the depth, if the actual depth is too difficult to obtain. Switch to a new location for the public web repository, which became necessary by a hardware problem. Fix the creation of symbolic links from a private data base to a public data base of cohomoloy rings. Fix comparison of an MTX matrix with None.
- v2.1.2: Cope with the new versions of Cython and Singular. Use Sage’s coercion model more properly. Build the documentation locally.
- v2.1.1: Usage of symbolic links to the public database, so that one can use (but of course not install) this package even without write permission in
`SAGE_DATA`

. Restructuring the code. Parallel testing is now only permitted if ticket #10004 is applied (September 2010).- v2.1: Support for big endian machines. 100% doctest coverage, parallel testing. New: Essential and depth essential ideals. Improved completion tests. (September 2010).
- v2.0: Cohomology of non prime power groups (April 2010).
- v1.2.p0: Improved test for the presence of the Small Groups library (thanks to Dima Pasechnik, March 2010).
- 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).
- v1.1: Added restricted Massey powers and general Massey products (August 2009).
- v1.0.2: Fixes some bugs (July 2009).
- v1.0.1: First public version in GPL 2 or later (July 2009)

## Licence¶

This document and our data bases of cohomology rings are licensed under a Creative Commons Attribution-Share Alike 3.0 License.

The code of our package is licensed under the GNU General Public License (GPL) version 2 or later, at your choice.

AUTHORS:

- Simon King <simon.king@uni-jena.de>
- David Green <david.green@uni-jena.de>