The HAPprime package for the GAP computer algebra system provides tools to compute mod-p cohomology rings of finite p-groups. It is an extension of the HAP package, upon which it depends.
Contents
1. Main features of the
HAPprime package
2. Example computations
3. Download and installation
4. Further information
5. Thanks
Main features of the HAPprime package
HAPprime provides two main components:- it improves on the existing HAP functions for mod-p cohomology rings of finite p-groups so that they work on slightly larger groups. This is achieved by more efficient memory use, with a choice of two methods which allow speed or memory use to be prioritised.
- it implements a tool for proving that the cohomology rings are correct. This is a an implementation of the Len Evens' original proof of the finite presentation of the cohomology rings, and uses Singular's Gröbner basis algorithms and HAP's implementation of CTC Wall's resolution of a group extension
Example computations
The following example uses HAPprime to compute the (provably-correct) mod-p cohomology ring for the dihedral group of order 64:
gap> G := DihedralGroup(64);; gap> ModPCohomologyRingPresentationSpectralSequence(G); [ GF(2)[x_1,x_2,x_3], [ x_1^2+x_1*x_2 ], [ 1, 1, 2 ] ] gap> StringTime(time); " 0:00:00.516" |
Download and Installation
To download this package, details of the package's requirements, and installation instructions, please see the download section of this website.
Further information
- The userguide for the HAPprime package
- The datatypes reference manual for the HAPprime package
- The HAP website
- The GAP website
Various other authors have software for computing mod-p cohomology rings of finite p-groups and have presented their results online:
- Jon Carlson performed computations using the MAGMA computer algebra system and his results are available online.
- David Green used custom C-code and presented his results here. He is currently running a project to calculate the mod-p cohomology rings of all groups of order 128.
- The CRIME package for GAP, by Marcus Bishop also calculates mod-p cohomology rings of finite p-groups. This uses broadly similar techniques to those used in HAPprime, but is less efficient.
Thanks
The HAPprime package for GAP is supported by a Marie Curie Transfer of Knowledge grant based at the Department of Mathematics, NUI Galway (MTKD-CT-2006-042685)
This package is now maintained by Graham Ellis, graham.ellis (at) nuigalway,ie