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1 Introduction

### 1 Introduction

The HAPprime package is a GAP package which supplements the HAP package (http://hamilton.nuigalway.ie/Hap/www/), providing new and improved functions for doing homological algebra over small prime-power groups. A detailed overview of the HAPprime package, with examples and documentation of the high-level functions, is provided in the accompanying HAPprime user guide.

This document, the datatypes reference manual, supplements the HAPprime user guide. It describes the new GAP datatypes defined by the HAPprime package, and all of the associated functions for working with each of these datatypes. The datatypes are

`FpGModuleGF`

  (Chapter 5) a free FG-module compactly represented in terms of generating elements, with operations that do as much manipulation as possible within this form, thus minimizing memory use.

`FpGModuleHomomorphismGF`

  (Chapter 6) a free linear homomorphism between two FG-modules, each represented as a `FpGModuleGF`. this also uses the compact generator form to save memory in its operations.

`HAPResolution`

  (Chapter 2) this datatype, defined in the HAP package, represents a free FG-resolution of a FG-module. HAPprime extends the definition of this datatype to save memory, and provides additional functions to operate on resolutions.

`HAPDerivation`

  (Chapter 8) a derivation over a polynomial ring R. In particular, HAPprime provides functions to calculate the kernel and homology of derivations for polynomials over prime fields.

In addition, Chapter 10 provides documentation for some general functions defined in HAPprime which extend some of the basic GAP functionality in areas such as matrices and polynomials.

Each chapter of this reference manual begins with an overview of the datatype, and then implementation details of any interesting functions. The function reference of related functions then follows, subdivided into sections of related functions. Examples demonstrating the use of each function are given at the end of each section.

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