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\Chapter{Introduction}

\atindex{Cubefree}{@{\Cubefree}}

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\Section{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
\cite{Hol93} investigated them at the end of the 19th century. Taunt
\cite{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 \cite{Di05} and \cite{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 \cite{BeEia}, \cite{BeEib}, and \cite{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 \cite{Di05} and \cite{DiEi05}. It is a refinement of
the methods of the {\GrpConst} package which are described in \cite{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 \cite{FlOB05} for which this package
also contains an implementation. Alternatively, the {\Irredsol} package
\cite{Irredsol} could be used for primes $p\le 251$.

The algorithm of \cite{FlOB05} requires a method to rewrite a matrix
representation. We use and implement the method of \cite{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.
