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# HAPprime

## More homological algebra with prime power groups

Version 0.3.1

16 December 2008

Paul Smith
Email: paul.smith@nuigalway.ie
Homepage: http://www.maths.nuigalway.ie/~pas
Address:
Department of Mathematics,
National University of Ireland, Galway
Galway,
Ireland.

### Copyright

© 2006-2008 Paul Smith

HAPprime is released under the GNU General Public License (GPL). This file is part of HAPprime, though as documentation it is released under the GNU Free Documentation License (see http://www.gnu.org/licenses/licenses.html#FDL).

HAPprime is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

HAPprime is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with HAPprime; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA

For more details, see http://www.fsf.org/licenses/gpl.html.

### Acknowledgements

HAPprime is supported by a Marie Curie Transfer of Knowledge grant based at the Department of Mathematics, NUI Galway (MTKD-CT-2006-042685)

### Contents

1 Introduction
1.1 Introduction to the HAPprime package
1.2 Required software
1.3 Installing HAPprime
1.4 Loading and testing HAPprime
1.5 Documentation
1.5-1 MakeHAPprimeDoc
1.6 Displaying progress and calculation information
1.6-1 InfoHAPprime
2 Examples
2.1 Computing the mod p group cohomology
2.2 Computing mod-p cohomology rings and their Poincaré series
2.2-1 A ring presentation for the mod p cohomology (up to degree n)
2.2-2 Calculating a provably-correct mod-p cohomology
2.2-3 Computing Poincaré series
2.3 Comparing the memory usage and speed of HAPprime and HAP's `ResolutionPrimePowerGroup` functions
2.3-1 HAPprime takes less memory to store resolutions
2.3-2 HAPprime takes less memory to compute resolutions
2.3-3 Automatic selection of the best method
3 Functions for Homological Algebra
3.1 Resolutions
3.1-1 ResolutionPrimePowerGroup
3.1-2 ExtendResolutionPrimePowerGroup
3.2 Poincaré Series
3.2-1 PoincareSeriesLHS
3.3 Cohomology Ring structure
3.3-1 ModPCohomologyRingPresentation
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