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#############################################################################
##
#W basicmat.gi GAP Library Frank Celler
##
#H @(#)$Id: basicmat.gi,v 4.10 2002/09/05 14:40:23 gap Exp $
##
#Y Copyright (C) 1996, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
##
## This file contains the methods for the construction of the basic matrix
## group types.
##
Revision.basicmat_gi :=
"@(#)$Id: basicmat.gi,v 4.10 2002/09/05 14:40:23 gap Exp $";
#############################################################################
##
#M CyclicGroupCons( <IsMatrixGroup>, <field>, <n> )
##
InstallOtherMethod( CyclicGroupCons,
"matrix group for given field",
true,
[ IsMatrixGroup and IsFinite,
IsField,
IsInt and IsPosRat ],
0,
function( filter, fld, n )
local o, m, i;
o := One(fld);
m := NullMat( n, n, fld );
for i in [ 1 .. n-1 ] do
m[i][i+1] := o;
od;
m[n][1] := o;
m := GroupByGenerators( [ ImmutableMatrix(fld,m) ] );
SetSize( m, n );
return m;
end );
#############################################################################
##
#M CyclicGroupCons( <IsMatrixGroup>, <n> )
##
InstallMethod( CyclicGroupCons,
"matrix group for default field",
true,
[ IsMatrixGroup and IsFinite,
IsInt and IsPosRat ],
0,
function( filter, n )
local m, i;
m := NullMat( n, n, Rationals );
for i in [ 1 .. n-1 ] do
m[i][i+1] := 1;
od;
m[n][1] := 1;
m := GroupByGenerators( [ ImmutableMatrix(Rationals,m) ] );
SetSize( m, n );
return m;
end );
#############################################################################
##
#M GeneralLinearGroupCons( <IsMatrixGroup>, <d>, <F> )
##
InstallMethod( GeneralLinearGroupCons,
"matrix group for dimension and finite field size",
[ IsMatrixGroup and IsFinite,
IsInt and IsPosRat,
IsField and IsFinite ],
function( filter, n, f )
local q, z, o, mat1, mat2, i, g;
q:= Size( f );
# small cases
if q = 2 and 1 < n then
return SL( n, 2 );
fi;
# construct the generators
z := PrimitiveRoot( f );
o := One( f );
mat1 := IdentityMat( n, o );
mat1[1][1] := z;
mat2 := List( Zero(o) * mat1, ShallowCopy );
mat2[1][1] := -o;
mat2[1][n] := o;
for i in [ 2 .. n ] do mat2[i][i-1]:= -o; od;
mat1 := ImmutableMatrix( f, mat1 );
mat2 := ImmutableMatrix( f, mat2 );
g := GroupByGenerators( [ mat1, mat2 ] );
SetName( g, Concatenation("GL(",String(n),",",String(q),")") );
SetDimensionOfMatrixGroup( g, n );
SetFieldOfMatrixGroup( g, f );
SetIsNaturalGL( g, true );
SetIsFinite(g,true);
if n<50 or n+q<500 then
Size(g);
fi;
# Return the group.
return g;
end );
#############################################################################
##
#M SpecialLinearGroupCons( <IsMatrixGroup>, <d>, <q> )
##
InstallMethod( SpecialLinearGroupCons,
"matrix group for dimension and finite field size",
[ IsMatrixGroup and IsFinite,
IsInt and IsPosRat,
IsField and IsFinite ],
function( filter, n, f )
local q, g, o, z, mat1, mat2, i, size, qi;
q:= Size( f );
# handle the trivial case first
if n = 1 then
g := GroupByGenerators( [ ImmutableMatrix( f, [[One(f)]] ) ] );
# now the general case
else
# construct the generators
o := One(f);
z := PrimitiveRoot(f);
mat1 := IdentityMat( n, o );
mat2 := List( Zero(o) * mat1, ShallowCopy );
mat2[1][n] := o;
for i in [ 2 .. n ] do mat2[i][i-1]:= -o; od;
if q = 2 or q = 3 then
mat1[1][2] := o;
else
mat1[1][1] := z;
mat1[2][2] := z^-1;
mat2[1][1] := -o;
fi;
mat1 := ImmutableMatrix(f,mat1);
mat2 := ImmutableMatrix(f,mat2);
g := GroupByGenerators( [ mat1, mat2 ] );
fi;
# set name, dimension and field
SetName( g, Concatenation("SL(",String(n),",",String(q),")") );
SetDimensionOfMatrixGroup( g, n );
SetFieldOfMatrixGroup( g, f );
SetIsFinite( g, true );
if q = 2 then
SetIsNaturalGL( g, true );
fi;
SetIsNaturalSL( g, true );
SetIsFinite(g,true);
# add the size
if n<50 or n+q<500 then
Size(g);
fi;
# return the group
return g;
end );
#############################################################################
##
#E
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