This chapter describes the methods available from the Cubefree package.
This section lists the implemented functions.
ConstructAllCFGroups(
order ) F
The input order has to be a positive cubefree integer. The output is a complete and irredundant list of isomorphism type representatives of groups of this size. If possible, the groups are given as pc groups and as permutations groups otherwise.
ConstructAllCFSolvableGroups(
order ) F
The input order has to be a positive cubefree integer. The output is a complete and irredundant list of isomorphism type representatives of solvable groups of this size. The groups are given as pc groups.
ConstructAllCFNilpotentGroups(
order ) F
The input order has to be a positive cubefree integer. The output is a complete and irredundant list of isomorphism type representatives of nilpotent groups of this size. The groups are given as pc groups.
ConstructAllCFSimpleGroups(
order ) F
The input order has to be a positive cubefree integer. The output is a complete and irredundant list of isomorphism type representatives of simple groups of this size. In particular, there exists either none or exactly one simple group of the given order.
ConstructAllCFFrattiniFreeGroups(
order ) F
The input order has to be a positive cubefree integer. The output is a complete and irredundant list of isomorphism type representatives of Frattini-free groups of this size.
NumberCFGroups(
n[,
bool ] ) F
The input n has to be a positive cubefree integer and the output is the number of all cubefree groups of order n. The SmallGroups library is used whenever possible, i.e. when nleq50000. Only if bool is set to false, then only the numbers of squarefree groups are taken from the SmallGroups library.
NumberCFSolvableGroups(
n[,
bool ] ) F
The input n has to be a positive cubefree integer and the output is the number of all cubefree solvable groups of order n. The SmallGroups library is used whenever possible, i.e. when nleq50000. Only if bool is set to false, then only the numbers of squarefree groups are taken from the SmallGroups library.
CountAllCFGroupsUpTo(
n[,
bool ]) F
The input is a positive integer n and the output is a list L of size n such that L[i] contains the number of isomorphism types of groups of order i if i is cubefree and L[i] is not bound, otherwise, 1leqi leqn. The SmallGroups library is used whenever possible, i.e. when nleq50000. Only if bool is set to false, then only the numbers of squarefree groups are taken from the SmallGroups library.
IsCubeFreeInt(
n ) P
The output is true if n is a cubefree integer and false otherwise.
IsSquareFreeInt(
n ) P
The output is true if n is a squarefree integer and false otherwise.
IrreducibleSubgroupsOfGL(
n,
q ) O
The current version of this function allows only n=2. The input q has to be a prime-power q=pr with pgeq5 a prime. The output is a list of all irreducible subgroups of GL(2,q) up to conjugacy.
RewriteAbsolutelyIrreducibleMatrixGroup(
G )
The input G has to be an absolutely irreducible matrix group over a finite field GF(q). If possible, the output is G rewritten over the subfield of GF(q) generated by the traces of the elements of G. If no rewriting is possible, then the input G is returned.
This section provides some useful information about the implementations.
ConstructAllCFGroups
The function ConstructAllCFGroups
constructs all groups of a given
cubefree order up to isomorphism using the Frattini Extension Method as described in Di05,
DiEi05, BeEia, and BeEib. One step in the Frattini
Extension Method is to compute Frattini extensions
and for this purpose some already implemented
methods of the required GAP package GrpConst are used.
Since ConstructAllCFGroups
requires only
some special types of irreducible subgroups of GL(2,p) (e.g. of cubefree order), it
contains a modified internal version of
IrreducibleSubgroupsOfGL
. This means that the latter is not called explicitely by
ConstructAllCFGroups
.
To reduce runtime, the generators of the reducible subgroups of GL(2,p), 2leqp leq100 a prime, are stored in the file 'diagonalMatrices.dat'.
ConstructAllCFSimpleGroups and ConstructAllCFNilpotentGroups
The construction of simple or nilpotent groups of cubefree order is rather easy, see Di05 or DiEi05. In particular, the methods used in these cases are independent from the methods used in the general cubefree case.
CountAllCFGroupsUpTo
As described in Di05 and DiEi05, every cubefree group G has
the form G=AtimesI where A is trivial or non-abelian simple and I is
solvable. Further, there is a one-to-one correspondence between the solvable
cubefree groups and some solvable Frattini-free groups. This one-to-one
correspondence allows to count the number of groups of a given cubefree order without
computing any Frattini extension.
To reduce runtime, the
computed irreducible and reducible subgroups of the general linear groups
GL(2,p) and also the number of the computed solvable
Frattini-free groups are stored during the whole computation. This is very
memory consuming but reduces the runtime significantly. The alternative is to run a loop over NumberCFGroups
.
IrreducibleSubgroupsOfGL
If the input is a matrix group over GF(q), then the algorithm needs to construct GF(q3) internally.
RewriteAbsolutelyIrreducibleMatrixGroup
The function RewriteAbsolutelyIrreducibleMatrixGroup
as described
algorithmically in
GlHo97 is probabilistic. If the input is GleqGL(d,pr), then the
expected runtime is O(rd3).
We have compared the results of ConstructAllCFGroups
with the library of
cubefree groups of GrpConst. Further, we compared the number and size of the
solvable groups constructed by IrreducibleSubgroupsOfGL
with the library of Irredsol.
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