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%W  intro.tex          Cubefree documentation              Heiko Dietrich
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%Y  2005
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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} package for constructing groups of cubefree order; that is, groups
whose order is not divisible by any third power of a prime.

The groups of squarefree order are known for a long time, since 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 up to isomorphism
all groups of a given cubefree order. 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 $p$. These subgroups are
constructed using the irreducible subgroups of GL$(2,p)$. To determine these
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 $p\le 251$.

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

For the construction of groups of squarefree order it is more practical to
use the efficient function <AllSmallGroups> of the {\GrpConst} package.

A more detailed describtion 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.
