1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2%% 3%W intro.tex Cubefree documentation Heiko Dietrich 4%% 5%% 6 7%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8\Chapter{Introduction} 9 10\atindex{Cubefree}{@{\Cubefree}} 11 12%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13\Section{Overview} 14 15This manual describes the {\Cubefree} package, 16a {\GAP} 4 package for constructing groups of cubefree order; i.e., groups 17whose order is not divisible by any third power of a prime. 18 19The groups of squarefree order are known for a long time: Hoelder 20\cite{Hol93} investigated them at the end of the 19th century. Taunt 21\cite{Tau55} has considered solvable groups of cubefree order, since he 22examined solvable groups with abelian Sylow subgroups. Cubefree groups in 23general are investigated firstly in \cite{Di05}, \cite{DiEi05}, and \cite{DiEi05add}, and this 24package contains the implementation of the algorithms described 25there. 26 27Some general approaches to construct groups of an arbitrarily given order are 28described in \cite{BeEia}, \cite{BeEib}, and \cite{BeEiO}. 29 30The main function of this package is a method to construct 31all groups of a given cubefree order up to isomorphism. The algorithm behind this function is 32described completely in \cite{Di05} and \cite{DiEi05}. It is a refinement of 33the methods of the {\GrpConst} package which are described in \cite{GrpConst}. 34 35This main function needs a method to construct up to conjugacy the solvable 36cubefree subgroups of GL$(2,p)$ coprime to $p$. We split this construction 37into the construction of reducible and irreducible subgroups of GL$(2,p)$. To determine the 38irreducible subgroups we use the method described in \cite{FlOB05} for which this package 39also contains an implementation. Alternatively, the {\IrredSol} package 40\cite{Irredsol} could be used for primes $p\le 251$. 41 42The algorithm of \cite{FlOB05} requires a method to rewrite a matrix 43representation. We use and implement the method of \cite{GlHo97} for this purpose. 44 45One can modify the construction algorithm for cubefree groups to a very 46efficient algorithm to construct groups of squarefree order. This is already 47done in the {\SmallGroups} library. Thus for the construction of groups of squarefree order it is more practical to 48use `AllSmallGroups' of the {\SmallGroups} library. 49 50A more detailed description of the implemented methods can be found in Chapter 2. 51 52Chapter "Installing and Loading the Cubefree Package" explains 53how to install and load the {\Cubefree} package. 54 55 56%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 57\Section{Theoretical background} 58 59In this section we give a brief survey about the main algorithm which is used 60to construct groups of cubefree order: the Frattini extension method. For a by 61far more detailed description 62we refer to the above references; e.g. see the online version of \cite{Di05}. 63 64Let $G$ be a finite group. The Frattini subgroup $\Phi(G)$ is defined to be 65the intersection of all maximal subgroups of $G$. We say a group $H$ is a 66Frattini extension by $G$ if the Frattini factor $H/\Phi(H)$ is isomorphic to 67$G$. The Frattini factor of $H$ is Frattini-free; i.e. it has a trivial 68Frattini subgroup. It is known that every prime divisor of $|H|$ is also a divisor of 69$|H/\Phi(H)|$. Thus the Frattini subgroup of a cubefree group has to be 70squarefree and, as it is nilpotent, it is a direct product of cyclic groups of 71prime order. 72 73Hence in order to construct all groups of a given cubefree order $n$, say, one can, 74firstly, construct all Frattini-free groups of suitable orders and, secondly, compute 75all corresponding Frattini extensions of order $n$. A first fundamental result 76is that a group of cubefree order is either a solvable Frattini 77extension or a direct product of a PSL$(2,r)$, $r>3$ a prime, with a solvable 78Frattini extension. In particular, the simple groups of cubefree 79order are the groups PSL$(2,r)$ with $r>3$ a prime such that $r\pm 801$ is cubefree. As a nilpotent group is the direct product of its Sylow subgroups, it 81is straightforward to compute all nilpotent groups of a given cubefree order. 82 83Another important result is 84that for a cubefree solvable Frattini-free group there is exactly one isomorphism 85type of suitable Frattini extensions, which restricts the construction of 86cubefree groups to the 87determination of cubefree solvable 88Frattini-free groups. This uniqueness of Frattini extensions is the 89main reason why the Frattini extension method works so efficiently in the 90cubefree case. 91 92In other words, there is a one-to-one correspondence between 93the solvable cubefree groups of order $n$ and {\it some} Frattini-free groups of order 94dividing $n$. This allows to count the number of isomorphism types of cubefree groups of a given 95order without 96constructing Frattini extensions. 97 98In the remaining part of this section we consider the construction of the 99solvable Frattini-free groups of a given cubefree order up to 100isomorphism. Such a group is a split extension over its socle; i.e. over the 101product of its minimal normal subgroups. Let $F$ be a solvable Frattini-free 102group of cubefree order with socle $S$. Then $S$ is a (cubefree) direct product of cyclic 103groups of prime order and $F$ can be written as $F=K\ltimes S$ 104where $K\leq$Aut$(S)$ is determined up to conjugacy. In particular, $K$ is a subdirect product of certain 105cubefree subgroups of groups of the type GL$(2,p)$ or 106$C_{p-1}$. Hence in order to determine all possible subgroups $K$ one can 107determine all possible projections from such a subgroup into the direct factors 108of the types GL$(2,p)$ and $C_{p-1}$, and then form all subdirect 109products having these projections. The construction of these subdirect 110products is one of the most time-consuming parts in the Frattini extension 111method for cubefree groups. 112