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#############################################################################
##
#W general.gi FORMAT Bettina Eick
#W conversion from GAP3 by C.R.B. Wright
##
#############################################################################
#M PPart( <elt>, <prime> ). . . . . . . . . . . . . . . . . . . . . . .local
## generator of <prime>-Sylow subgroup of < <elt> >
InstallMethod( PPart, "generic unnormalized", true,
[IsMultiplicativeElementWithInverse, IsPosInt], 0,
function(g, prime)
local o, facs, x;
if not IsPrimeInt( prime ) then
Error( prime, " must be a prime in PPart.");
fi;
o := Order(g);
if o = 1 then return g; fi;
facs := Factors(o);
x := Length(Filtered(facs, y -> y = prime));
return g^(o/(prime^x)); # no gcd normalization as in PrimePowerComponent
end);
#############################################################################
#M PPrimePart( <elt>, <prime> ). . . . . . . . . . . . . . . . . . . . local
## generator of <prime>-complement of < <elt> >
InstallMethod( PPrimePart, "generic unnormalized", true,
[IsMultiplicativeElementWithInverse, IsPosInt], 0,
function(g, prime)
local o, facs, x;
if not IsPrimeInt( prime ) then
Error( prime, " must be a prime in PPrimePart.");
fi;
o := Order(g);
if o = 1 then return g; fi;
facs := Factors(o);
x := Length(Filtered(facs, y -> y = prime));
return g^(prime^x); # no gcd normalization
end);
#############################################################################
#M PiPrimePart( <elt>, <list of primes> ). . . . . . . . . . . . . . . local
## generator of pi-complement of < <elt> >
InstallMethod( PiPrimePart, "generic unnormalized", true,
[IsMultiplicativeElementWithInverse, IsList], 0,
function( g, pi )
local p, o, facs, exp;
for p in pi do
if not IsPrimeInt(p) then
Error(pi," is not a list of primes.\n");
fi;
od;
o := Order(g);
if o = 1 then return g; fi;
facs := Factors(o);
exp := Product(Filtered(facs, p -> p in pi));
return g^exp;
end);
#############################################################################
#F ( G, <pcgsR>, <pcgsL>, <pcgsK> ) . . . . . . . . . . local
## find pcgs of [<L>, <R>] * <K>
InstallGlobalFunction(FCommutatorPcgs,
function( G, pcgsR, pcgsL, pcgsK )
local K, gens, Rigs, C, r, g, c, spg;
## It is assumed that L/K is abelian.
K := SubgroupByPcgs( G, pcgsK );
if Length( pcgsK ) >0 then
gens := pcgsL mod pcgsK;
else
gens := pcgsL;
fi;
Rigs := pcgsR mod pcgsL;
C := [];
for r in Rigs do
for g in gens do
c := Comm(r, g);
if not c in K then
AddSet(C, c);
fi;
od;
od;
if C = [] then return pcgsK; fi;
spg := SpecialPcgs(G);
## normalized 4-20-00. Without it, system.base may not be normalized
## and FExponents will hang in a loop.
return NormalizedPcgs( spg,
InducedPcgsByPcSequenceAndGenerators( spg, pcgsK, C) );
end);
#############################################################################
#F FCentralTest( G, <pcgsR>, <pcgsL>, <pcgsK> ) . . . . . . . . .local
## check [<L>, <R>] <= <K>
InstallGlobalFunction(FCentralTest,
function( G, pcgsR, pcgsL, pcgsK )
local K, gens, Rigs, r, g, c;
K := SubgroupByPcgs( G, pcgsK );
gens := pcgsL mod pcgsK;
Rigs := pcgsR mod pcgsL;
for r in Rigs do
for g in gens do
c := Comm(r, g);
if not c in K then
return false;
fi;
od;
od;
return true;
end);
#############################################################################
#F FExponents( <system>, <elt>, <field>, <int>, <int> ) . . . collector info
##
InstallGlobalFunction(FExponents,
function( system, u, F, first, next )
local sph, depths, expo, d, i, tmp, max, j;
# catch trivial case
if Length( system.base ) = 0 then
return [];
fi;
# set up
sph := SpecialPcgs( system.H ); ## may be known already as system.sph
max := Length( system.base );
depths := system.depths{[first..max]};
expo := List( [first..max], x -> 0 );
# iterate
d := DepthOfPcElement( sph, u );
i := Position( depths, d );
while not IsBool(i) do
tmp := LeadingExponentOfPcElement( sph, u );
expo[ i ] := tmp;
u := LeftQuotient( system.base[first+i-1]^tmp , u );
d := DepthOfPcElement( sph, u );
i := Position( depths, d );
od;
# cut tail off
expo := expo{[1..next-first+1]};
# throw it into the correct field
if IsFFE( One( F )) then
return expo * One( F );
elif IsInt( One( F ) ) then
return expo;
else
Error(One( F ), " seems not to be a field element ");;
fi;
end);
#############################################################################
#M GpByNiceMonomorphism( <nice>, <group> ) construct group with nice obj
## Same as undocumented GroupByNiceMonomorphism from the library.
InstallMethod( GpByNiceMonomorphism,
true,
[ IsGroupHomomorphism,
IsGroup ],
0,
function( nice, grp )
local fam, pre;
fam := FamilyObj( Source(nice) );
pre := Objectify(NewType(fam,IsGroup and IsAttributeStoringRep), rec());
SetIsHandledByNiceMonomorphism( pre, true );
SetNiceMonomorphism( pre, nice );
SetNiceObject( pre, grp );
SetOne(pre,One(Source(nice)));
UseIsomorphismRelation(grp,pre);
return pre;
end );
#############################################################################
#F SubgpMethodByNiceMonomorphism( <oper>, <par> )
## Same as undocumented SubgroupMethodByNiceMonomorphism from the library.
BindGlobal( "SubgpMethodByNiceMonomorphism", function( oper, par )
# check the argument length
if 1 <> Length(par) then
Error( "need only one argument for ", NameFunction(oper) );
fi;
par := ShallowCopy(par);
par[1] := par[1] and IsHandledByNiceMonomorphism;
# install the method
InstallOtherMethod( oper,
"handled by nice monomorphism",
true,
par,
0,
function( obj )
local nice, img, sub;
nice := NiceMonomorphism(obj);
img := oper( NiceObject(obj) );
sub := GpByNiceMonomorphism( nice, img );
SetParent( sub, obj );
return sub;
end );
end );
#############################################################################
#F SubgpMethodByNiceMonomorphismCollOther( <oper>, <par> )
## Same as undocumented SubgroupMethodByNiceMonomorphismCollOther.
BindGlobal( "SubgpMethodByNiceMonomorphismCollOther",
function( oper, par )
# check the argument length
if 2 <> Length(par) then
Error( "need two arguments for ", NameFunction(oper) );
fi;
par := ShallowCopy(par);
par[1] := par[1] and IsHandledByNiceMonomorphism;
# install the method
InstallOtherMethod( oper,
"handled by nice monomorphism",
true,
par,
0,
function( obj, other )
local nice, img, sub;
nice := NiceMonomorphism(obj);
img := oper( NiceObject(obj), other );
sub := GpByNiceMonomorphism( nice, img );
SetParent( sub, obj );
return sub;
end );
end );
#E End of general.gi
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