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# 1 Introduction

### Sections

1. Overview and Background

## 1.1 Overview and Background

This manual describes the Cubefree package, a GAP 4 package for constructing groups of cubefree order; i.e., groups whose order is not divisible by any third power of a prime.

The groups of squarefree order are known for a long time: Hoelder Hol93 investigated them at the end of the 19th century. Taunt Tau55 has considered solvable groups of cubefree order, since he examined solvable groups with abelian Sylow subgroups. Cubefree groups in general are investigated firstly in Di05 and DiEi05, and this package contains the implementation of the algorithms described there.

Some general approaches to construct groups of an arbitrarily given order are described in BeEia, BeEib, and BeEiO.

The main function of this package is a method to construct all groups of a given cubefree order up to isomorphism. The algorithm behind this function is described completely in Di05 and DiEi05. It is a refinement of the methods of the GrpConst package which are described in GrpConst.

This main function needs a method to construct up to conjugacy the solvable cubefree subgroups of GL(2,p) coprime to p. We split this construction into the construction of reducible and irreducible subgroups of GL(2,p). To determine the irreducible subgroups we use the method described in FlOB05 for which this package also contains an implementation. Alternatively, the Irredsol package Irredsol could be used for primes ple251.

The algorithm of FlOB05 requires a method to rewrite a matrix representation. We use and implement the method of GlHo97 for this purpose.

One can modify the construction algorithm for cubefree groups to a very efficient algorithm to construct groups of squarefree order. This is already done in the GrpConst package. Thus for the construction of groups of squarefree order it is more practical to use `AllSmallGroups` of the GrpConst package.

A more detailed description of the implemented methods can be found in Chapter 2.

Chapter Installing and Loading the Cubefree Package explains how to install and load the Cubefree package.

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