%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %W intro.tex Cubefree documentation Heiko Dietrich %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Introduction} \atindex{Cubefree}{@{\Cubefree}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Overview} 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 {\SmallGroups} library. Thus for the construction of groups of squarefree order it is more practical to use `AllSmallGroups' of the {\SmallGroups} library. 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Theoretical background} In this section we give a brief survey about the main algorithm which is used to construct groups of cubefree order: the Frattini extension method. For a by far more detailed description we refer to the above references; e.g. see the online version of \cite{Di05}. Let $G$ be a finite group. The Frattini subgroup $\Phi(G)$ is defined to be the intersection of all maximal subgroups of $G$. We say a group $H$ is a Frattini extension by $G$ if the Frattini factor $H/\Phi(H)$ is isomorphic to $G$. The Frattini factor of $H$ is Frattini-free; i.e. it has a trivial Frattini subgroup. It is known that every prime divisor of $|H|$ is also a divisor of $|H/\Phi(H)|$. Thus the Frattini subgroup of a cubefree group has to be squarefree and, as it is nilpotent, it is a direct product of cyclic groups of prime order. Hence in order to construct all groups of a given cubefree order $n$, say, one can, firstly, construct all Frattini-free groups of suitable orders and, secondly, compute all corresponding Frattini extensions of order $n$. A first fundamental result is that a group of cubefree order is either a solvable Frattini extension or a direct product of a PSL$(2,r)$, $r>3$ a prime, with a solvable Frattini extension. In particular, the simple groups of cubefree order are the groups PSL$(2,r)$ with $r>3$ a prime such that $r\pm 1$ is cubefree. As a nilpotent group is the direct product of its Sylow subgroups, it is straightforward to compute all nilpotent groups of a given cubefree order. Another important result is that for a cubefree solvable Frattini-free group there is exactly one isomorphism type of suitable Frattini extensions, which restricts the construction of cubefree groups to the determination of cubefree solvable Frattini-free groups. This uniqueness of Frattini extensions is the main reason why the Frattini extension method works so efficiently in the cubefree case. In other words, there is a one-to-one correspondence between the solvable cubefree groups of order $n$ and {\it some} Frattini-free groups of order dividing $n$. This allows to count the number of isomorphism types of cubefree groups of a given order without constructing Frattini extensions. In the remaining part of this section we consider the construction of the solvable Frattini-free groups of a given cubefree order up to isomorphism. Such a group is a split extension over its socle; i.e. over the product of its minimal normal subgroups. Let $F$ be a solvable Frattini-free group of cubefree order with socle $S$. Then $S$ is a (cubefree) direct product of cyclic groups of prime order and $F$ can be written as $F=K\ltimes S$ where $K\leq$Aut$(S)$ is determined up to conjugacy. In particular, $K$ is a subdirect product of certain cubefree subgroups of groups of the type GL$(2,p)$ or $C_{p-1}$. Hence in order to determine all possible subgroups $K$ one can determine all possible projections from such a subgroup into the direct factors of the types GL$(2,p)$ and $C_{p-1}$, and then form all subdirect products having these projections. The construction of these subdirect products is one of the most time-consuming parts in the Frattini extension method for cubefree groups.