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#############################################################################
##
#W glzmodmz.gi GAP library Stefan Kohl
##
#H @(#)$Id: glzmodmz.gi,v 1.2 2002/09/05 14:42:08 gap Exp $
##
#Y (C) 2001 School Math and Comp. Sci., University of St. Andrews, Scotland
##
Revision.glzmodmz_gi :=
"@(#)$Id: glzmodmz.gi,v 1.2 2002/09/05 14:42:08 gap Exp $";
#############################################################################
##
#F SizeOfGLdZmodmZ( <d>, <m> ) . . . . . . Size of the group GL(<d>,Z/<m>Z)
##
## Computes the order of the group `GL( <d>, Integers mod <m> )' for
## positive integers <d> and <m> > 1.
##
InstallGlobalFunction( SizeOfGLdZmodmZ,
function ( d, m )
local size, pow, p, q, k, i;
if not (IsPosInt(d) and IsInt(m) and m > 1)
then Error("GL(",d,",Integers mod ",m,") is not a well-defined group, ",
"resp. not supported.\n");
fi;
size := 1;
for pow in Collected(Factors(m)) do
p := pow[1]; k := pow[2]; q := p^k;
size := size * Product([d*k - d .. d*k - 1], i -> q^d - p^i);
od;
return size;
end );
#############################################################################
##
#M SpecialLinearGroupCons( IsNaturalSL, <d>, Integers mod <m> )
##
InstallMethod( SpecialLinearGroupCons,
"natural SL for dimension and residue class ring",
[ IsMatrixGroup and IsFinite, IsPosInt,
IsRing and IsFinite and IsZmodnZObjNonprimeCollection ],
function ( filter, d, R )
local G, gens, g, m, T;
m := Size(R);
if R <> Integers mod m or m = 1 then TryNextMethod(); fi;
if IsPrime(m) then return SpecialLinearGroupCons(IsMatrixGroup,d,m); fi;
if d = 1
then gens := [IdentityMat(d,R)];
else gens := List(GeneratorsOfGroup(SymmetricGroup(d)),
g -> PermutationMat(g,d) * One(R));
for g in gens do
if DeterminantMat(g) <> One(R) then g[1] := -g[1]; fi;
od;
T := IdentityMat(d,R); T[1][2] := One(R); Add(gens,T);
fi;
G := GroupByGenerators(gens);
SetName(G,Concatenation("SL(",String(d),",Z/",String(m),"Z)"));
SetIsNaturalSL(G,true);
SetDimensionOfMatrixGroup(G,d);
SetIsFinite(G,true);
SetSize(G,SizeOfGLdZmodmZ(d,m)/Phi(m));
return G;
end );
#############################################################################
##
#M GeneralLinearGroupCons( IsNaturalGL, <d>, Integers mod <m> )
##
InstallMethod( GeneralLinearGroupCons,
"natural GL for dimension and residue class ring",
[ IsMatrixGroup and IsFinite, IsPosInt,
IsRing and IsFinite and IsZmodnZObjNonprimeCollection ],
function ( filter, d, R )
local G, gens, g, m, T, D;
m := Size(R);
if R <> Integers mod m or m = 1 then TryNextMethod(); fi;
if IsPrime(m) then return GeneralLinearGroupCons(IsMatrixGroup,d,m); fi;
if d = 1
then gens := List(GeneratorsOfGroup(Units(R)), g -> [[g]]);
else gens := List(GeneratorsOfGroup(SymmetricGroup(d)),
g -> PermutationMat(g,d) * One(R));
T := IdentityMat(d,R); T[1][2] := One(R); Add(gens,T);
for g in GeneratorsOfGroup(Units(R)) do
D := IdentityMat(d,R); D[1][1] := g; Add(gens,D);
od;
fi;
G := GroupByGenerators(gens);
SetName(G,Concatenation("GL(",String(d),",Z/",String(m),"Z)"));
SetIsNaturalGL(G,true);
SetDimensionOfMatrixGroup(G,d);
SetIsFinite(G,true);
SetSize(G,SizeOfGLdZmodmZ(d,m));
return G;
end );
#############################################################################
##
#E glzmodmz.gi . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
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