<html><head><title>[cubefree] 1 Introduction</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>1 Introduction</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP001.htm#SECT001">Overview</a> <li> <A HREF="CHAP001.htm#SECT002">Theoretical background</a> </ol><p> <p> <a name = "I0"></a> <p> <h2><a name="SECT001">1.1 Overview</a></h2> <p><p> This manual describes the <font face="Gill Sans,Helvetica,Arial">Cubefree</font> package, a <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package for constructing groups of cubefree order; i.e., groups whose order is not divisible by any third power of a prime. <p> The groups of squarefree order are known for a long time: Hoelder <a href="biblio.htm#Hol93"><cite>Hol93</cite></a> investigated them at the end of the 19th century. Taunt <a href="biblio.htm#Tau55"><cite>Tau55</cite></a> has considered solvable groups of cubefree order, since he examined solvable groups with abelian Sylow subgroups. Cubefree groups in general are investigated firstly in <a href="biblio.htm#Di05"><cite>Di05</cite></a> and <a href="biblio.htm#DiEi05"><cite>DiEi05</cite></a>, and this package contains the implementation of the algorithms described there. <p> Some general approaches to construct groups of an arbitrarily given order are described in <a href="biblio.htm#BeEia"><cite>BeEia</cite></a>, <a href="biblio.htm#BeEib"><cite>BeEib</cite></a>, and <a href="biblio.htm#BeEiO"><cite>BeEiO</cite></a>. <p> 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 <a href="biblio.htm#Di05"><cite>Di05</cite></a> and <a href="biblio.htm#DiEi05"><cite>DiEi05</cite></a>. It is a refinement of the methods of the <font face="Gill Sans,Helvetica,Arial">GrpConst</font> package which are described in <a href="biblio.htm#GrpConst"><cite>GrpConst</cite></a>. <p> This main function needs a method to construct up to conjugacy the solvable cubefree subgroups of GL<var>(2,p)</var> coprime to <var>p</var>. We split this construction into the construction of reducible and irreducible subgroups of GL<var>(2,p)</var>. To determine the irreducible subgroups we use the method described in <a href="biblio.htm#FlOB05"><cite>FlOB05</cite></a> for which this package also contains an implementation. Alternatively, the <font face="Gill Sans,Helvetica,Arial">IrredSol</font> package <a href="biblio.htm#Irredsol"><cite>Irredsol</cite></a> could be used for primes <var>ple251</var>. <p> The algorithm of <a href="biblio.htm#FlOB05"><cite>FlOB05</cite></a> requires a method to rewrite a matrix representation. We use and implement the method of <a href="biblio.htm#GlHo97"><cite>GlHo97</cite></a> for this purpose. <p> 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 <font face="Gill Sans,Helvetica,Arial">SmallGroups</font> library. Thus for the construction of groups of squarefree order it is more practical to use <code>AllSmallGroups</code> of the <font face="Gill Sans,Helvetica,Arial">SmallGroups</font> library. <p> A more detailed description of the implemented methods can be found in Chapter 2. <p> Chapter <a href="CHAP003.htm">Installing and Loading the Cubefree Package</a> explains how to install and load the <font face="Gill Sans,Helvetica,Arial">Cubefree</font> package. <p> <p> <h2><a name="SECT002">1.2 Theoretical background</a></h2> <p><p> 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 <a href="biblio.htm#Di05"><cite>Di05</cite></a>. <p> Let <var>G</var> be a finite group. The Frattini subgroup <var>Phi(G)</var> is defined to be the intersection of all maximal subgroups of <var>G</var>. We say a group <var>H</var> is a Frattini extension by <var>G</var> if the Frattini factor <var>H/Phi(H)</var> is isomorphic to <var>G</var>. The Frattini factor of <var>H</var> is Frattini-free; i.e. it has a trivial Frattini subgroup. It is known that every prime divisor of <var>|H|</var> is also a divisor of <var>|H/Phi(H)|</var>. 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. <p> Hence in order to construct all groups of a given cubefree order <var>n</var>, say, one can, firstly, construct all Frattini-free groups of suitable orders and, secondly, compute all corresponding Frattini extensions of order <var>n</var>. A first fundamental result is that a group of cubefree order is either a solvable Frattini extension or a direct product of a PSL<var>(2,r)</var>, <var>r>3</var> a prime, with a solvable Frattini extension. In particular, the simple groups of cubefree order are the groups PSL<var>(2,r)</var> with <var>r>3</var> a prime such that <var>rpm 1</var> 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. <p> 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. <p> In other words, there is a one-to-one correspondence between the solvable cubefree groups of order <var>n</var> and <em>some</em>Frattini-free groups of order dividing <var>n</var>. This allows to count the number of isomorphism types of cubefree groups of a given order without constructing Frattini extensions. <p> 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 <var>F</var> be a solvable Frattini-free group of cubefree order with socle <var>S</var>. Then <var>S</var> is a (cubefree) direct product of cyclic groups of prime order and <var>F</var> can be written as <var>F=KltimesS</var> where <var>Kleq</var>Aut<var>(S)</var> is determined up to conjugacy. In particular, <var>K</var> is a subdirect product of certain cubefree subgroups of groups of the type GL<var>(2,p)</var> or <var>C<sub>p-1</sub></var>. Hence in order to determine all possible subgroups <var>K</var> one can determine all possible projections from such a subgroup into the direct factors of the types GL<var>(2,p)</var> and <var>C<sub>p-1</sub></var>, 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. <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>cubefree manual<br>October 2007 </address></body></html>
