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3%W  intro.tex          Cubefree documentation              Heiko Dietrich
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8\Chapter{Introduction}
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10\atindex{Cubefree}{@{\Cubefree}}
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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}.
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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.
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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