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#############################################################################
##
#W basic.gd GAP Library Frank Celler
##
#H @(#)$Id: basic.gd,v 4.28 2000/02/01 13:34:11 gap Exp $
##
#Y Copyright (C) 1996, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
##
## This file contains the operations for the construction of the basic group
## types.
##
Revision.basic_gd :=
"@(#)$Id: basic.gd,v 4.28 2000/02/01 13:34:11 gap Exp $";
#############################################################################
##
#1
## There are several infinite families of groups which are parametrized by
## numbers.
## {\GAP} provides various functions to construct these groups.
## The functions always permit (but do not require) one to indicate
## a filter (see~"Filters"), for example `IsPermGroup', `IsMatrixGroup' or
## `IsPcGroup', in which the group shall be constructed.
## There always is a default filter corresponding to a ``natural'' way
## to describe the group in question.
## Note that not every group can be constructed in every filter,
## there may be theoretical restrictions (`IsPcGroup' only works for
## solvable groups) or methods may be available only for a few filters.
##
## Certain filters may admit additional hints.
## For example, groups constructed in `IsMatrixGroup' may be constructed
## over a specified field, which can be given as second argument of the
## function that constructs the group;
## The default field is `Rationals'.
#############################################################################
##
#O TrivialGroupCons( <filter> )
##
DeclareConstructor( "TrivialGroupCons", [ IsGroup ] );
#############################################################################
##
#F TrivialGroup( [<filter>] ) . . . . . . . . . . . . . . . . trivial group
##
## constructs a trivial group in the category given by the filter <filter>.
## If <filter> is not given it defaults to `IsPcGroup'.
##
BindGlobal( "TrivialGroup", function( arg )
if Length( arg ) = 0 then
return TrivialGroupCons( IsPcGroup );
elif IsFilter( arg[1] ) and Length( arg ) = 1 then
return TrivialGroupCons( arg[1] );
fi;
Error( "usage: TrivialGroup( [<filter>] )" );
end );
#############################################################################
##
#O AbelianGroupCons( <filter>, <ints> )
##
DeclareConstructor( "AbelianGroupCons", [ IsGroup, IsList ] );
#############################################################################
##
#F AbelianGroup( [<filt>, ]<ints> ) . . . . . . . . . . . . . abelian group
##
## constructs an abelian group in the category given by the filter <filt>
## which is of isomorphism type $C_{ints[1]} \* C_{ints[2]} \* \ldots \*
## C_{ints[n]}$. <ints> must be a list of positive integers. If <filt> is
## not given it defaults to `IsPcGroup'. The generators of the group
## returned are the elements corresponding to the integers in <ints>.
##
BindGlobal( "AbelianGroup", function ( arg )
if Length(arg) = 1 then
return AbelianGroupCons( IsPcGroup, arg[1] );
elif IsOperation(arg[1]) then
if Length(arg) = 2 then
return AbelianGroupCons( arg[1], arg[2] );
elif Length(arg) = 3 then
return AbelianGroupCons( arg[1], arg[2], arg[3] );
fi;
fi;
Error( "usage: AbelianGroup( [<filter>, ]<ints> )" );
end );
#############################################################################
##
#O AlternatingGroupCons( <filter>, <deg> )
##
DeclareConstructor( "AlternatingGroupCons", [ IsGroup, IsInt ] );
#############################################################################
##
#F AlternatingGroup( [<filt>, ]<deg> ) . . . . . . . . . . alternating group
#F AlternatingGroup( [<filt>, ]<dom> ) . . . . . . . . . . alternating group
##
## constructs the alternating group of degree <deg> in the category given
## by the filter <filt>.
## If <filt> is not given it defaults to `IsPermGroup'.
## In the second version, the function constructs the alternating group on
## the points given in the set <dom> which must be a set of positive
## integers.
##
BindGlobal( "AlternatingGroup", function ( arg )
if Length(arg) = 1 then
return AlternatingGroupCons( IsPermGroup, arg[1] );
elif IsOperation(arg[1]) then
if Length(arg) = 2 then
return AlternatingGroupCons( arg[1], arg[2] );
fi;
fi;
Error( "usage: AlternatingGroup( [<filter>, ]<deg> )" );
end );
#############################################################################
##
#O CyclicGroupCons( <filter>, <n> )
##
DeclareConstructor( "CyclicGroupCons", [ IsGroup, IsInt ] );
#############################################################################
##
#F CyclicGroup( [<filt>, ]<n> ) . . . . . . . . . . . . . . . cyclic group
##
## constructs the cyclic group of size <n> in the category given by the
## filter <filt>. If <filt> is not given it defaults to `IsPcGroup'.
##
BindGlobal( "CyclicGroup", function ( arg )
if Length(arg) = 1 then
return CyclicGroupCons( IsPcGroup, arg[1] );
elif IsOperation(arg[1]) then
if Length(arg) = 2 then
return CyclicGroupCons( arg[1], arg[2] );
elif Length(arg) = 3 then
return CyclicGroupCons( arg[1], arg[2], arg[3] );
fi;
fi;
Error( "usage: CyclicGroup( [<filter>, ]<size> )" );
end );
#############################################################################
##
#O DihedralGroupCons( <filter>, <n> )
##
DeclareConstructor( "DihedralGroupCons", [ IsGroup, IsInt ] );
#############################################################################
##
#F DihedralGroup( [<filt>, ]<n> ) . . . . . . . dihedral group of order <n>
##
## constructs the dihedral group of size <n> in the category given by the
## filter <filt>. If <filt> is not given it defaults to `IsPcGroup'.
##
BindGlobal( "DihedralGroup", function ( arg )
if Length(arg) = 1 then
return DihedralGroupCons( IsPcGroup, arg[1] );
elif IsOperation(arg[1]) then
if Length(arg) = 2 then
return DihedralGroupCons( arg[1], arg[2] );
elif Length(arg) = 3 then
return DihedralGroupCons( arg[1], arg[2], arg[3] );
fi;
fi;
Error( "usage: DihedralGroup( [<filter>, ]<size> )" );
end );
#############################################################################
##
#O ElementaryAbelianGroupCons( <filter>, <n> )
##
DeclareConstructor( "ElementaryAbelianGroupCons", [ IsGroup, IsInt ] );
#############################################################################
##
#F ElementaryAbelianGroup( [<filt>, ]<n> ) . . . . elementary abelian group
##
## constructs the elementary abelian group of size <n> in the category
## given by the filter <filt>.
## If <filt> is not given it defaults to `IsPcGroup'.
##
BindGlobal( "ElementaryAbelianGroup", function ( arg )
if Length(arg) = 1 then
return ElementaryAbelianGroupCons( IsPcGroup, arg[1] );
elif IsOperation(arg[1]) then
if Length(arg) = 2 then
return ElementaryAbelianGroupCons( arg[1], arg[2] );
elif Length(arg) = 3 then
return ElementaryAbelianGroupCons( arg[1], arg[2], arg[3] );
fi;
fi;
Error( "usage: ElementaryAbelianGroup( [<filter>, ]<size> )" );
end );
#############################################################################
##
#O ExtraspecialGroupCons( <filter>, <order>, <exponent> )
##
DeclareConstructor( "ExtraspecialGroupCons", [ IsGroup, IsInt, IsObject ] );
#############################################################################
##
#F ExtraspecialGroup( [<filt>, ]<order>, <exp> ) . . . . extraspecial group
##
## Let <order> be of the form $p^{2n+1}$, for a prime integer $p$ and a
## positive integer $n$.
## `ExtraspecialGroup' returns the extraspecial group of order <order>
## that is determined by <exp>, in the category given by the filter <filt>.
##
## If $p$ is odd then admissible values of <exp> are the exponent of the
## group (either $p$ or $p^2$) or one of `{'}+{'}', `\"+\"', `{'}-{'}',
## `\"-\"'.
## For $p = 2$, only the above plus or minus signs are admissible.
##
## If <filt> is not given it defaults to `IsPcGroup'.
##
BindGlobal( "ExtraspecialGroup", function ( arg )
if Length(arg) = 2 then
return ExtraspecialGroupCons( IsPcGroup, arg[1], arg[2] );
elif IsOperation(arg[1]) then
if Length(arg) = 3 then
return ExtraspecialGroupCons( arg[1], arg[2], arg[3] );
elif Length(arg) = 4 then
return ExtraspecialGroupCons( arg[1], arg[2], arg[3], arg[4] );
fi;
fi;
Error( "usage: ExtraspecialGroup( [<filter>, ]<order>, <exponent> )" );
end );
#############################################################################
##
#O MathieuGroupCons( <filter>, <degree> )
##
DeclareConstructor( "MathieuGroupCons", [ IsGroup, IsInt ] );
#############################################################################
##
#F MathieuGroup( [<filt>, ]<degree> ) . . . . . . . . . . . . Mathieu group
##
## constructs the Mathieu group of degree <degree> in the category given by
## the filter <filt>,
## where <degree> must be in $\{ 9, 10, 11, 12, 21, 22, 23, 24 \}$.
## If <filt> is not given it defaults to `IsPermGroup'.
##
BindGlobal( "MathieuGroup", function( arg )
if Length( arg ) = 1 then
return MathieuGroupCons( IsPermGroup, arg[1] );
elif IsOperation( arg[1] ) then
if Length( arg ) = 2 then
return MathieuGroupCons( arg[1], arg[2] );
elif Length( arg ) = 3 then
return MathieuGroupCons( arg[1], arg[2], arg[3] );
fi;
fi;
Error( "usage: MathieuGroup( [<filter>, ]<degree> )" );
end );
#############################################################################
##
#O SymmetricGroupCons( <filter>, <deg> )
##
DeclareConstructor( "SymmetricGroupCons", [ IsGroup, IsInt ] );
#############################################################################
##
#F SymmetricGroup( [<filt>, ]<deg> )
#F SymmetricGroup( [<filt>, ]<dom> )
##
## constructs the symmetric group of degree <deg> in the category given by
## the filter <filt>.
## If <filt> is not given it defaults to `IsPermGroup'.
## In the second version, the function constructs the symmetric group on
## the points given in the set <dom> which must be a set of positive
## integers.
##
BindGlobal( "SymmetricGroup", function ( arg )
if Length(arg) = 1 then
return SymmetricGroupCons( IsPermGroup, arg[1] );
elif IsOperation(arg[1]) then
if Length(arg) = 2 then
return SymmetricGroupCons( arg[1], arg[2] );
fi;
fi;
Error( "usage: SymmetricGroup( [<filter>, ]<deg> )" );
end );
#############################################################################
##
#E
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