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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346#############################################################################
##
#W claspcgs.gi GAP library Heiko Theißen
##
##
#Y Copyright (C) 1997, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains functions that deal with conjugacy topics in solvable
## groups using affine methods. These topics includes calculating the
## (rational) conjugacy classes and centralizers in solvable groups. The
## functions rely only on the existence of pcgs, not on the particular
## representation of the groups.
##
#############################################################################
##
#F SubspaceVectorSpaceGroup( <N>, <p>, <gens>, <howmuch> )
##
## This function creates a record containing information about a complement
## in <N> to the span of <gens>.
##
InstallGlobalFunction( SubspaceVectorSpaceGroup, function( N, p, gens,howmuch )
local zero, one, r, ran, n, nan, cg, pos, Q, i, j, v;
one:=One( GF( p ) ); zero:=0 * one;
r:=Length( N ); ran:=[ 1 .. r ];
n:=Length( gens ); nan:=[ 1 .. n ];
Q:=[ ];
if n <> 0 and IsMultiplicativeElementWithInverse( gens[ 1 ] ) then
Q:=List( gens, gen -> ExponentsOfPcElement( N, gen ) ) * one;
else
Q:=ShallowCopy( gens );
fi;
cg:=rec( matrix :=[ ],
one :=one,
baseComplement:=ShallowCopy( ran ),
commutator :=0,
centralizer :=0,
dimensionN :=r,
dimensionC :=n );
if n = 0 or r = 0 then
cg.inverse:=NullMapMatrix;
cg.projection :=IdentityMat( r, one );
cg.needed :=[];
return cg;
fi;
for i in nan do
cg.matrix[ i ]:=Concatenation( Q[ i ], zero * nan );
cg.matrix[ i ][ r + i ]:=one;
od;
TriangulizeMat( cg.matrix );
pos:=1;
for v in cg.matrix do
while v[ pos ] = zero do
pos:=pos + 1;
od;
RemoveSet( cg.baseComplement, pos );
if pos <= r then cg.commutator :=cg.commutator + 1;
else cg.centralizer:=cg.centralizer + 1; fi;
od;
if howmuch=1 then
return Immutable(cg);
fi;
cg.needed :=[ ];
cg.projection :=IdentityMat( r, one );
# Find a right pseudo inverse for <Q>.
Append( Q, cg.projection );
Q:=MutableTransposedMat( Q );
TriangulizeMat( Q );
Q:=TransposedMat( Q );
i:=1;
j:=1;
while i <= Length( N ) do
while j <= Length( gens ) and Q[ j ][ i ] = zero do
j:=j + 1;
od;
if j <= Length( gens ) and Q[ j ][ i ] <> zero then
cg.needed[ i ]:=j;
else
# If <Q> does not have full rank, terminate when the bottom row
# is reached.
i:=Length( N );
fi;
i:=i + 1;
od;
if IsEmpty( cg.needed ) then
cg.inverse:=NullMapMatrix;
else
cg.inverse:=Q{ Length( gens ) + ran }
{ [ 1 .. Length( cg.needed ) ] };
cg.inverse:=ImmutableMatrix(p,cg.inverse,true);
fi;
if IsEmpty( cg.baseComplement ) then
cg.projection:=NullMapMatrix;
else
# Find a base change matrix for the projection onto the complement.
for i in [ 1 .. cg.commutator ] do
cg.projection[ i ][ i ]:=zero;
od;
Q:=[ ];
for i in [ 1 .. cg.commutator ] do
Q[ i ]:=cg.matrix[ i ]{ ran };
od;
for i in [ cg.commutator + 1 .. r ] do
Q[ i ]:=ListWithIdenticalEntries( r, zero );
Q[ i ][ cg.baseComplement[ i-r+Length(cg.baseComplement) ] ]
:=one;
od;
cg.projection:=cg.projection ^ Q;
cg.projection:=cg.projection{ ran }{ cg.baseComplement };
cg.projection:=ImmutableMatrix(p,cg.projection,true);
fi;
return Immutable(cg);
end );
#############################################################################
##
#F KernelHcommaC( <N>, <h>, <C>, <howmuch> )
##
## Given a homomorphism C -> N, c |-> [h,c], this function determines (a) a
## vector space decomposition N = [h,C] + K with projection onto K and (b)
## the ``kernel'' S < C which plays the role of C_G(h) in lemma 3.1 of
## [Mecky, Neub\"user, Bull. Aust. Math. Soc. 40].
##
InstallGlobalFunction( KernelHcommaC, function( N, h, C, howmuch )
local i, tmp, v,x;
x:=List( C, c -> Comm( h, c ) );
N!.subspace:=SubspaceVectorSpaceGroup(N,RelativeOrders(N)[1],x,howmuch);
tmp:=[ ];
for i in [ N!.subspace.commutator + 1 ..
N!.subspace.commutator + N!.subspace.centralizer ] do
v:=N!.subspace.matrix[ i ];
tmp[ i - N!.subspace.commutator ]:=PcElementByExponentsNC( C,
v{ [ N!.subspace.dimensionN + 1 ..
N!.subspace.dimensionN + N!.subspace.dimensionC ] } );
od;
return tmp;
end );
#############################################################################
##
#F CentralStepClEANS( <homepcgs>,<H>, <U>, <N>, <cl>,<off> )
##
# if <off> is true the normal subgroup is not necessarily in the series and
# we cannot call `ExtendedPcgs' but must form a new pcgs.
InstallGlobalFunction( CentralStepClEANS, function( home,H, U, N, cl,off )
local classes, # classes to be constructed, the result
field, # field over which <N> is a vector space
h, # preimage `cl.representative' under <hom>
gens, # preimage `Centralizer( cl )' under <hom>
cemodk,
cengen,
exp, w, # coefficient vectors for projection along $[h,N]$
kern,img,
c,nc; # loop variable
field:=GF( RelativeOrders( N )[ 1 ] );
h:=cl.representative;
if IsBound(cl.centralizerpcgs) then
if IsSubset(cl.centralizerpcgs,DenominatorOfModuloPcgs(N!.capH)) then
cemodk:=Filtered(cl.centralizerpcgs,i->not i in
DenominatorOfModuloPcgs(N!.capH));
else
cemodk:=cl.centralizerpcgs mod
DenominatorOfModuloPcgs( N!.capH );
fi;
else
cemodk:=InducedPcgs(home, cl.centralizer ) mod
DenominatorOfModuloPcgs( N!.capH );
fi;
kern:=DenominatorOfModuloPcgs( N!.capH );
if IsBound(cl.candidates) then
img:=KernelHcommaC( N, h, cemodk,2 );
else
img:=KernelHcommaC( N, h, cemodk,1 );
fi;
if off then
cengen:=InducedPcgsByPcSequenceAndGenerators(ParentPcgs( kern ),
kern, img );
else
#cengen:=ExtendedPcgs(kern,img);
cengen:=Concatenation(img,kern);
fi;
#C:=SubgroupByPcgs( H, cengen );
classes:=[ ];
if IsBound( cl.candidates ) then
gens:=cemodk{ N!.subspace.needed };
if IsIdenticalObj( FamilyObj( U ), FamilyObj( cl.candidates ) ) then
for c in cl.candidates do
exp:=ExponentsOfPcElement( N, LeftQuotient( h, c ) );
MultRowVector( exp, One( field ) );
w:=exp * N!.subspace.projection;
exp{ N!.subspace.baseComplement }:=
exp{ N!.subspace.baseComplement }-w;
nc:=rec( representative:=h * PcElementByExponentsNC
( N, N!.subspace.baseComplement, w ),
#centralizer:=C,
#centralizerpcgs:=cengen,
cengen:=cengen,
operator:=LinearCombinationPcgs( gens,
exp * N!.subspace.inverse,
One( cl.candidates[1] ))^(-1));
# check that action is really OK
Assert(1,c^nc.operator/nc.representative in
Group(DenominatorOfModuloPcgs(N),One(U)));
Add( classes, nc );
od;
else
c:=rec( representative:=cl.candidates,
#centralizer:=C,
#centralizerpcgs:=cengen,
cengen:=cengen,
operator:=One( H ) );
Add( classes, c );
fi;
else
gens:=N!.subspace.baseComplement;
for w in field ^ Length( gens ) do
c:=rec( representative:=h * PcElementByExponentsNC( N,gens,w ),
#centralizer:=C )
#centralizerpcgs:=cengen )
cengen:=cengen );
Add( classes, c );
od;
fi;
return classes;
end );
#############################################################################
##
#F CorrectConjugacyClass(<home>,<h>,<n>,<stabpcgs>,<N>,<depth>,<cNh>,<off> )
## cf. MN89
##
InstallGlobalFunction( CorrectConjugacyClass,
function( home, h, n, stab, N,depthlev, cNh,off )
local cl, comm, s, ostab;
ostab:=stab;
#AH: take only those elements module N - the part in N is cNh
stab:=Filtered(stab,i->DepthOfPcElement(home,i)<depthlev);
if Length(N!.subspace.inverse)>0 and Length(stab)>0 then
comm:=[];
for s in [ 1 .. Length( stab ) ] do
comm[ s ]:=ExponentsOfPcElement( N,
Comm( n, stab[ s ] )*Comm( h, stab[ s ] ));
od;
comm:=comm * N!.subspace.inverse;
for s in [ 1 .. Length( comm ) ] do
stab[ s ]:=stab[ s ] / PcElementByExponentsNC
( N!.capH, N!.subspace.needed, comm[ s ] );
od;
fi;
if off then
stab:=InducedPcgsByPcSequenceAndGenerators(ParentPcgs( cNh ),
cNh, stab );
elif IsList(cNh) and IsList(cNh[1]) then
#stab:=ExtendedPcgs(cNh[1],Concatenation(stab,cNh[2]));
stab:=Concatenation(stab,cNh[2],cNh[1]);
else
#stab:=ExtendedPcgs(cNh,stab);
stab:=Concatenation(stab,cNh);
fi;
cl:=rec( representative:=h * n,
cengen:=stab );
return cl;
end );
#############################################################################
##
#F GeneralStepClEANS( <homepcgs>, <H>, <U>, <N>,<nexpo>, <cl>,<off> )
##
# if <off> is true the normal subgroup is not necessarily in the series and
# we cannot call `ExtendedPcgs' but must form a new pcgs.
InstallGlobalFunction(GeneralStepClEANS,function(home, H, U, N,nexpo, cl, off)
local classes, # classes to be constructed, the result
field, # field over which <N> is a vector space
h, # preimage `cl.representative' under <hom>
cNh, # centralizer of <h> in <N>
gens, # preimage `Centralizer( cl )' under <hom>
r, # dimension of <N>
ran, # constant range `[ 1 .. r ]'
aff, # <N> as affine space
xset, # affine operation of <C> on <aff>
imgs, M, # generating matrices for affine operation
orb, # orbit of affine operation
Rep, # representative function to use for <orb>
n, k, # cf. Mecky--Neub\"user paper
cls,rep,pos,# set of classes with canonical representatives
j,
c, ca, i, # loop variables
S, # orbit-stabilizer
ceve,#xponent vector
p, # positions
Cgens, # generators of C in N
next,blist, # orbit stabilizer algo
depthlev, # depth at which N starts
one,zero,
vec,
kern,img;
depthlev:=DepthOfPcElement(home,N[1]);
Cgens:=cl.centralizerpcgs;
field:=GF( RelativeOrders( N )[ 1 ] );
h:=cl.representative;
# Determine the subspace $[h,N]$ and calculate the centralizer of <h>.
kern:=DenominatorOfModuloPcgs( N!.capH );
img:=KernelHcommaC( N, h, N!.capH,2 );
r:=Length( N!.subspace.baseComplement );
#AH: Take only those which are not in N
gens:=Cgens mod NumeratorOfModuloPcgs(N!.capH);
if not (off or IsBound(cl.candidates)) and r=0 then
# special treatment: The commutators span the whole space
# this is noncentral_case4 in GAP3
c:=CorrectConjugacyClass( home, h, One(gens[1]),
gens, N, depthlev,[kern,img],off );
return [c];
fi;
ran:=[ 1 .. r ];
if off then
cNh:=InducedPcgsByPcSequenceAndGenerators(ParentPcgs( kern ),
kern, img );
else
#cNh:=ExtendedPcgs(kern,img);
cNh:=[kern,img]; # we only need cNh to extend it
fi;
# Construct matrices for the affine operation on $N/[h,N]$.
aff:=ExtendedVectors( field ^ r );
one:=One(field);
zero:=Zero(field);
imgs:=[ ];
for c in gens do
ceve:=ExponentsOfPcElement(home,c,[1..depthlev-1]);
M:=[ ];
for i in [ 1 .. r ] do
p:=N!.subspace.baseComplement[i];
# construct the vector image
vec:=p;
for j in [1..Length(ceve)] do
for k in [1..ceve[j]] do
if IsInt(vec) then
vec:=nexpo[j][vec];
else
vec:=vec*nexpo[j];
fi;
od;
od;
M[ i ]:=Concatenation( vec
* N!.subspace.projection, [ zero ] );
od;
i:=Comm( h, c );
M[ r + 1 ]:=Concatenation( ExponentsOfPcElement
( N, i ) * N!.subspace.projection,
[ one ] );
M:=ImmutableMatrix(field,M,true);
Add( imgs, M );
od;
classes:=[ ];
if IsBound( cl.candidates ) then
# not yet improved: we use an external set and thus have to give a
# full list of generators of C:
imgs:=Concatenation(imgs,List([1..Length(Cgens)-Length(gens)],i->IdentityMat( r + 1, field )));
gens:=Cgens;
xset:=ExternalSet(SubgroupByPcgs(H,Cgens),aff,gens,imgs,OnPoints);
if IsIdenticalObj( FamilyObj( U ), FamilyObj( cl.candidates ) ) then
Rep:=CanonicalRepresentativeOfExternalSet;
else
cl.candidates:=[ cl.candidates ];
Rep:=Representative;
fi;
cls:=[ ];
for ca in cl.candidates do
n:=ExponentsOfPcElement( N, LeftQuotient( h, ca ) ) *
One( field );
ConvertToVectorRep(n, field);
k:=n * N!.subspace.projection;
orb:=Concatenation( k, [ One( field ) ]);
ConvertToVectorRep(orb, field);
orb:=ExternalOrbit( xset, Immutable(orb) );
rep:=PcElementByExponentsNC( N, N!.subspace.baseComplement,
Rep( orb ){ ran } );
pos:=Position( cls, rep );
if pos = fail then
Add( cls, rep );
c:=StabilizerOfExternalSet( orb );
if IsIdenticalObj( Rep, CanonicalRepresentativeOfExternalSet )
then
c:=ConjugateSubgroup( c, ActorOfExternalSet( orb ) );
fi;
c:=CorrectConjugacyClass( home, h, rep, InducedPcgs(home,c), N,
depthlev,cNh,off );
else
c:=rec( representative:=h * rep,
#centralizer:=classes[ pos ].centralizer )
#centralizerpcgs:=classes[ pos ].centralizerpcgs )
cengen:=classes[ pos ].cengen );
fi;
n:=ShallowCopy( -n );
n{ N!.subspace.baseComplement }:=
k + n{ N!.subspace.baseComplement };
c.operator:=PcElementByExponentsNC( N, N!.subspace.needed,
n * N!.subspace.inverse );
# Now (h.n)^c.operator = h.k
if IsIdenticalObj(Rep,CanonicalRepresentativeOfExternalSet) then
c.operator:=c.operator * ActorOfExternalSet( orb );
# Now (h.n)^c.operator = h.rep mod [h,N]
k:=PcElementByExponentsNC( N, N!.subspace.needed,
ExponentsOfPcElement( N, LeftQuotient
( c.representative, ca ^ c.operator ) ) *
N!.subspace.inverse );
c.operator:=c.operator / k;
# Now (h.n)^c.operator = h.rep
fi;
Add( classes, c );
od;
else
#xset:=ExternalSet( C, aff, gens, imgs );
#k:=ExternalOrbitsStabilizers( xset );
# do the orbits stuff ourselves
blist:=BlistList([1..Length(aff)],[]);
next:=1;
k:=[];
while next<>fail do
S:=Pcs_OrbitStabilizer(gens,aff,aff[next],imgs,OnRight);
# tick off
if IsPositionDictionary(S.dictionary) then
UniteBlist(blist,S.dictionary!.blist);
else
for i in S.orbit do
blist[PositionCanonical(aff,i)]:=true;
od;
fi;
Unbind(S.dictionary);
Add(k,S);
next:=Position(blist,false,next);
od;
for orb in k do
rep:=PcElementByExponentsNC( N, N!.subspace.baseComplement,
orb.orbit[1]{ ran } );
c:=CorrectConjugacyClass( home, h, rep,
#orb.stabilizer, N, depthlev,cNh,off )
orb.stabpcs, N, depthlev,cNh,off );
Add( classes, c );
od;
fi;
return classes;
end );
#############################################################################
##
#F ClassesSolvableGroup(<G>, <mode> [,<opt>]) . . . . .
##
## In this function classes are described by records with components
## `representative', `centralizer', `galoisGroup' (for rational classes). If
## <candidates> are given, their classes will have a canonical
## `representative'
##
InstallGlobalFunction(ClassesSolvableGroup, function(arg)
local G, home, # the group and the home pcgs
H,Hp, # acting group
mustlift,
liftkerns,
QH,QG,
fhome,ofhome,
first,
mode, # LSB: ratCl | power | test :MSB
candidates, # candidates to be replaced by their canonical reps.
eas, # elementary abelian series in <G>
step, # counter looping over <eas>
K, L, # members of <eas>
indstep, # indice normal steps
Ldep, # depth of L in pcgs
Kp,mK,Lp,mL, # induced and modulo pcgs's
LcapH,KcapH, # intersections
N, cent, # elementary abelian factor, for affine action
cls, newcls, # classes in range/source of homomorphism
cli, # index
news, # new classes obtained in step
cl, # class looping over <cls>
opr, exp, # (candidates[i]^opr[i])^exp[i]=cls[i].representative
team, # team of candidates with same image modulo <K>
blist,pos,q, # these control grouping of <cls> into <team>s
p, # prime dividing $|G|$
i,c, # loop variables
opt, # options
consider, # consider function
divi,
inflev, # InfoLevel flag
nexpo, # N-Exponents of the elements of N conjugated
allcent; # DivisorsInt(Size(G)) (used for Info)
inflev:=InfoLevel(InfoClasses)>1;
mode:=arg[2]; # explained below whenever it appears
if mode mod 2=1 then
Error("this function does not cater for rational classes any longer");
fi;
G:=arg[1];
if Length(arg)=3 then
opt:=ShallowCopy(arg[3]);
# convert series to pcgs
if IsBound(opt.series) and not IsBound(opt.pcgs) then
fi;
else
opt:=rec();
fi;
# <candidates> is a list of elements whose classes will be output (but
# with canonical representatives), see comment above. Or <candidates> is
# just one element, from whose output class the centralizer will be read
# off.
H:=G;
if IsBound(opt.candidates) then
candidates:=opt.candidates;
if not ForAll(candidates,i->i in G) then
G:=ClosureGroup(H,candidates);
fi;
else
candidates:=false;
fi;
if IsBound(opt.consider) then
consider:=opt.consider;
else
consider:=true;
fi;
# Treat the case of a trivial group.
if IsTrivial(H) then
if mode=4 then # test conjugacy of two elements
return One(G);
else
cl:=rec(representative:=One(G),
centralizer:=H);
fi;
if candidates<>false then
cls:=List(candidates, c -> cl);
else
cls:=[cl];
fi;
return cls;
fi;
# Calculate a (central) elementary abelian series with all pcgs induced
# w.r.t. <homepcgs>.
if IsBound(opt.pcgs) then
# we prescribed a series
home:=opt.pcgs;
eas:=EANormalSeriesByPcgs(home);
cent:=false;
elif IsPrimePowerInt(Size(G)) then
p:=FactorsInt(Size(G))[1];
home:=PcgsPCentralSeriesPGroup(G);
eas:=PCentralNormalSeriesByPcgsPGroup(home);
cent:=ReturnTrue;
else
home:=PcgsElementaryAbelianSeries(G);
eas:=EANormalSeriesByPcgs(home);
cent:=function(cl, N, L)
return ForAll(N, k -> ForAll
#(InducedPcgs(home,cl.centralizer),
(cl.centralizerpcgs,
#T was: Only those elements form the induced PCGS. The subset seemed to
#T enforce taking only the elements up, but the ordering of the series used
#T may be different then the ordering in the PCGS. So this will fail. AH
#T one might pick the right ones, but this would be almost the same work.
#T { [1 .. Length(InducedPcgsWrtHomePcgs(cl.centralizer))
#T - Length(InducedPcgsWrtHomePcgs(L))] },
c -> Comm(k, c) in L));
end;
cent:=false;
fi;
if cent=false then
# AH, 26-4-99: Test centrality not via `in' but via exponents
cent:=function(pcgs,grpg,Npcgs,dep)
local i,j;
for i in grpg do
for j in Npcgs do
if DepthOfPcElement(pcgs,Comm(j,i))<dep then
return false;
fi;
od;
od;
return true;
end;
fi;
indstep:=IndicesEANormalSteps(home);
# check to which factors we want to lift
mustlift:=List(eas,i->false);
liftkerns:=[];
if candidates=false then
# we only want to go in factor groups if no candidates are given
# (otherwise we'd have to take care not to forget tails when mapping in
# the factor groups)
step:=2; # the first step we'd have
for i in [2..Length(eas)-1] do
if Index(G,eas[i])>1000 or Index(G,eas[i+1])>10000 then
# only form a factor if the factor is large enough or the next step
# would be large
# form a factor by i and go to this factor at the first time (index
# step) no factor representation was given
mustlift[step]:=true;
liftkerns[step]:=eas[i];
step:=i+1;
fi;
od;
if step>2 then
# we created a factor, so we have to lift at the end
mustlift[step]:=true;
liftkerns[step]:=eas[Length(eas)];
fi;
fi;
Info(InfoClasses,1,"Series of sizes ",List(eas,Size));
if mode<3 and inflev then
divi:=DivisorsInt(Size(G));
Info(InfoClasses,2,"centsiz: ",divi);
fi;
# Initialize the algorithm for the trivial group.
step:=1;
L:=eas[step];
Lp:=InducedPcgs(home,L);
if not IsIdenticalObj( G, H ) then
Hp:=InducedPcgs(home, H );
LcapH:=NormalIntersectionPcgs( home, Hp, Lp );
fi;
if candidates<>false then
mL:=ModuloPcgsByPcSequenceNC(home, home, Lp);
fi;
cl:=rec(representative:=One(G),
centralizer:=H,
centralizerpcgs:=InducedPcgs(home,H),
cengen:=InducedPcgs(home,H));
if candidates<>false then
cls:=List(candidates, c -> cl);
opr:=List(candidates, c -> One(G));
exp:=ListWithIdenticalEntries(Length(candidates), 1);
else
cls:=[cl];
fi;
# Now go back through the factors by all groups in the elementary abelian
# series.
first:=true;
fhome:=home; # just to avoid unboundness the first time
QG:=G;
QH:=H;
for step in [step + 1 .. Length(eas)] do
Info(InfoClasses,1,"Step ",step,", ",Length(cls)," classes to lift");
# We apply the homomorphism principle to the homomorphism G/L -> G/K.
if mustlift[step] then
ofhome:=fhome;
# get the new quotient and Q's
if Size(eas[step])=1 then
QH:=H;
fhome:=home;
QG:=G;
else
# the new factor group in which we calculate
QH:=home mod InducedPcgs(home,liftkerns[step]);
QH:=GROUP_BY_PCGS_FINITE_ORDERS(QH);
fhome:=FamilyPcgs(QH);
QG:=SubgroupByPcgs(QG,
ProjectedInducedPcgs(home,fhome,InducedPcgs(home,G)));
fi;
fi;
# The actual computations are all done in <G>, factors are
# represented by modulo pcgs.
Ldep:=indstep[step];
if IsIdenticalObj(fhome,home) then
K:=eas[step-1];
Kp:=InducedPcgs(fhome,K);
L:=eas[step];
Lp:=InducedPcgs(fhome,L);
elif mustlift[step] then
Kp:=ProjectedInducedPcgs(home,fhome,InducedPcgs(home,eas[step-1]));
K:=SubgroupByPcgs(QG,Kp);
Lp:=ProjectedInducedPcgs(home,fhome,InducedPcgs(home,eas[step]));
L:=SubgroupByPcgs(QG,Lp); # not needed any longer
else
# we did not lift
K:=L;
Kp:=Lp;
Lp:=ProjectedInducedPcgs(home,fhome,InducedPcgs(home,eas[step]));
L:=SubgroupByPcgs(QG,Lp); # not needed any longer
fi;
N:=Kp mod Lp; # modulo pcgs representing the kernel
if mustlift[step] then
for i in cls do
if not IsBound(i.yet) then
if first then
# if it is the first time, we must actually map in the factor
i.representative:=ProjectedPcElement(home,fhome,i.representative);
i.centralizerpcgs:=ProjectedInducedPcgs(home,fhome,i.cengen);
i.cengen:=i.centralizerpcgs!.pcSequence;
else
i.representative:=LiftedPcElement(fhome,ofhome,i.representative);
i.centralizerpcgs:=LiftedInducedPcgs(fhome,ofhome,i.cengen,N);
i.cengen:=i.centralizerpcgs!.pcSequence;
fi;
i.yet:=true; # several cl records may be equal. We must map only
# once
fi;
od;
else
for i in cls do
if IsBound(i.cengen) and not IsBound(i.centralizerpcgs) then
i.centralizerpcgs:=InducedPcgsByPcSequence(fhome,i.cengen);
i.cengen:=i.centralizerpcgs!.pcSequence;
fi;
od;
fi;
first:=false;
# allcent:=ForAll(N,i->ForAll(GeneratorsOfGroup(G),j->Comm(i,j) in L))
allcent:=cent(fhome,fhome,N,Ldep);
if allcent=false then
nexpo:=LinearOperationLayer(fhome{[1..indstep[step-1]-1]},N);
fi;
#T What is this? Obviously it is needed somewhere, but it is
#T certainly not good programming style. AH
#SetFilterObj(N, IsPcgs);
if not IsIdenticalObj(G,H) then
Error("This case disabled -- code not yet corrected");
KcapH:=LcapH;
LcapH:=NormalIntersectionPcgs(fhome,Hp,Lp);
N!.capH:=KcapH mod LcapH;
SetFilterObj( N!.capH, IsPcgs );
else
N!.capH:=N;
fi;
# Identification of classes.
# Rational classes or identification of classes.
if candidates<>false then
mK:=mL;
mL:=ModuloPcgsByPcSequenceNC(fhome, fhome, Lp);
if mode=4 # test conjugacy of two elements
and not cls[1].representative /
cls[2].representative in K then
return fail;
fi;
blist:=BlistList([1 .. Length(cls)], []);
pos:=Position(blist, false);
while pos<>fail do
# Find a team of candidates with same image under <modK>.
cl:=cls[pos];
cl.representative:=PcElementByExponentsNC(mK,
ExponentsOfPcElement(mK, cl.representative));
cl.candidates:=[];
team:=[];
q:=pos;
while q<>fail do
if cls[q].representative /
cl.representative in K then
c:=candidates[q] ^ opr[q];
i:=PositionSorted(cl.candidates, c);
if i > Length(cl.candidates)
or cl.candidates[i]<>c then
Add(cl.candidates, c,i);
Add(team, [q], i);
else
Add(team[i], q);
fi;
blist[q]:=true;
fi;
q:=Position(blist, false, q);
od;
# Now <cl> is a class modulo <K> (possibly with
# `<cl>.candidates' a list of elements mapping into this
# class modulo <K>). Let <newcls> be a list of all classes
# modulo <L> that map to <cl> modulo <K> (resp. a list of
# classes to which the list `<cl>.candidates' maps modulo
# <K>, together with `operator's and `exponent's as in
# (c^o^e=r)).
if allcent then
# generic central
Info(InfoClasses,5,"central case 1");
newcls:=CentralStepClEANS(fhome,QH, QG, N, cl,false);
elif cent(fhome,cl.centralizerpcgs, N, Ldep) then
# central in this case
Info(InfoClasses,5,"central case 2");
newcls:=CentralStepClEANS(fhome,QH, QG, N, cl,false);
else
Info(InfoClasses,5,"general case");
newcls:=GeneralStepClEANS(fhome, QH, QG, N, nexpo, cl,false);
fi;
# Update <cls>, <opr> and <exp>.
for i in [1 .. Length(team)] do
for q in team[i] do
cls[q]:=newcls[i];
opr[q]:=opr[q] * newcls[i].operator;
od;
od;
pos:=Position(blist, false, pos);
od;
else
newcls:=[];
for cli in [1..Length(cls)] do
cl:=cls[cli];
if consider=true
or consider(fhome,cl.representative,cl.centralizerpcgs,K,L)
then
if allcent or cent(fhome,cl.centralizerpcgs, N, Ldep) then
news:=CentralStepClEANS(fhome,QG, QG, N, cl,false);
else
news:=GeneralStepClEANS(fhome, QG, QG, N,nexpo, cl,false);
fi;
Assert(1,# only do the test if no factors were formed
FamilyObj(news[1].cengen)<>FamilyObj(eas[step]) or
ForAll(news,
i->ForAll(i.cengen,
j->Comm(i.representative,j) in eas[step])));
Append(newcls,news);
fi;
Unbind(cls[cli]);
od;
cls:=newcls;
fi;
if inflev then
c:=Collected(List(cls,i->Size(SubgroupByPcgs(QH,
InducedPcgsByPcSequence(fhome,i.cengen)))));
if not IsBound( divi ) then
divi:=DivisorsInt(Size(G));
fi;
c:=Concatenation(c,List(divi,i->[i,0])); # to cope with `First'
Info(InfoClasses,2,List(divi,i->First(c,j->j[1]=i)[2]));
fi;
od;
if mode=4 then # test conjugacy of two elements
if cls[1].representative<>cls[2].representative then
return fail;
else
return opr[1] / opr[2];
fi;
fi;
for i in cls do
if not IsBound(i.centralizer) then
if not IsBound(i.centralizerpcgs) then
i.centralizerpcgs:=InducedPcgsByPcSequence(home,i.cengen);
i.cengen:=i.centralizerpcgs;
fi;
i.centralizer:=SubgroupByPcgs(G,i.centralizerpcgs);
fi;
od;
if candidates<>false then # add operators (and exponents)
for i in [1 .. Length(cls)] do
cls[i].operator:=opr[i];
od;
fi;
return cls;
end);
InstallGlobalFunction(CentralizerSizeLimitConsiderFunction,function(sz)
return function(fhome,rep,cenp,K,L)
return Product(RelativeOrders(cenp))/Size(K)<=sz;
end;
end);
#############################################################################
##
#M ActorOfExternalSet( <cl> ) . . . . . . . . . conj. cl. of solv. groups
##
InstallMethod( ActorOfExternalSet, true,
[ IsConjugacyClassGroupRep ], 0,
function( cl )
local G, rep;
G:=ActingDomain( cl );
if not CanEasilyComputePcgs( G ) then
TryNextMethod();
fi;
rep:=ClassesSolvableGroup( G, 0,rec(candidates:=[ Representative(cl)]) )
[ 1 ];
if not HasStabilizerOfExternalSet( cl ) then
SetStabilizerOfExternalSet( cl,
ConjugateSubgroup( rep.centralizer, rep.operator ^ -1 ) );
fi;
SetCanonicalRepresentativeOfExternalSet( cl, rep.representative );
return rep.operator;
end );
#############################################################################
#############################################################################
# everything which follows is only used for rational classes in p groups.
# This is not of that much importance any longer as the permutation groups
# class algorithm is different, but it is still worth having for rational
# classes of p-elements. AH, 14-apr-99
#############################################################################
##
#F RationalClassesSolvableGroup(<G>, <mode> [,<opt>]) . . . . .
##
## This is the old version. It is now only used for rational classes and
## does not incorporate any of the improvements to the ordinary code.
## (However therefore the ordinary code does not need to worry with the
## rational classes case)
## In this function classes are described by records with components
## `representative', `centralizer', `galoisGroup' (for rational classes). If
## <candidates> are given, their classes will have a canonical
## `representative'
## and additional components `operator' and `exponent' (for
## rational classes) such that
## (candidate ^ operator) ^ exponent=representative. (c^o^e=r)
##
InstallGlobalFunction(RationalClassesSolvableGroup, function(arg)
local G, home, # the group and the home pcgs
H,Hp, # acting group
mode, # LSB: ratCl | power | test :MSB
candidates, # candidates to be replaced by their canonical reps.
eas, # elementary abelian series in <G>
step, # counter looping over <eas>
K, L, # members of <eas>
Kp,mK,Lp,mL, # induced and modulo pcgs's
LcapH,KcapH, # intersections
N, cent, # elementary abelian factor, for affine action
cls, newcls, # classes in range/source of homomorphism
news, # new classes obtained in step
cl, # class looping over <cls>
opr, exp, # (candidates[i]^opr[i])^exp[i]=cls[i].representative
team, # team of candidates with same image modulo <K>
blist,pos,q, # these control grouping of <cls> into <team>s
p, # prime dividing $|G|$
ord, # order of a rational class modulo <L>
new, power, # auxiliary variables for determination of power tree
c, i, # loop variables
opt, # options
divi; # DivisorsInt(Size(G)) (used for Info)
G:=arg[1];
mode :=arg[2]; # explained below whenever it appears
if Length(arg)=3 then
opt:=ShallowCopy(arg[3]);
# convert series to pcgs
if IsBound(opt.series) and not IsBound(opt.pcgs) then
Error("convert series to pcgs!");
fi;
else
opt:=rec();
fi;
# <candidates> is a list of elements whose classes will be output (but
# with canonical representatives), see comment above. Or <candidates> is
# just one element, from whose output class the centralizer will be read
# off.
H:=G;
if IsBound(opt.candidates) then
candidates:=opt.candidates;
if not ForAll(candidates,i->i in G) then
G:=ClosureGroup(H,candidates);
fi;
else
candidates:=false;
fi;
#if IsBound(opt.consider) then
# consider:=opt.consider;
#else
# consider:=true;
#fi;
# Treat the case of a trivial group.
if IsTrivial(H) then
if mode=4 then # test conjugacy of two elements
return One(G);
elif mode mod 2=1 then # rational classes
cl:=rec(representative:=One(G),
centralizer:=G,
galoisGroup:=GroupByPrimeResidues([], 1));
cl.galoisGroup!.type:=3;
cl.galoisGroup!.operators:=[];
cl.isCentral:=true;
if mode mod 4=3 then # construct the power tree
cl.power :=rec(representative:=One(G));
cl.power.operator:=One(G);
cl.power.exponent:=1;
fi;
else
cl:=rec(representative:=One(G),
centralizer:=H);
fi;
if candidates<>false then
cls:=List(candidates, c -> cl);
else
cls:=[cl];
fi;
return cls;
fi;
# Calculate a (central) elementary abelian series with all pcgs induced
# w.r.t. <homepcgs>.
if IsBound(opt.pcgs) then
# we prescribed a series
home:=opt.pcgs;
eas:=EANormalSeriesByPcgs(home);
cent:=function(cl, N, L)
return ForAll(N, k -> ForAll
(InducedPcgs(home,cl.centralizer), c -> Comm(k, c) in L));
end;
elif IsPrimePowerInt(Size(G)) then
p:=FactorsInt(Size(G))[1];
home:=PcgsPCentralSeriesPGroup(G);
eas:=PCentralNormalSeriesByPcgsPGroup(home);
cent:=ReturnTrue;
elif mode mod 2=1 then # rational classes
Error("<G> must be a p-group");
else
home:=PcgsElementaryAbelianSeries(G);
eas:=EANormalSeriesByPcgs(home);
cent:=function(cl, N, L)
return ForAll(N, k -> ForAll
(InducedPcgs(home,cl.centralizer),
#T was: Only those elements form the induced PCGS. The subset seemed to
#T enforce taking only the elements up, but the ordering of the series used
#T may be different then the ordering in the PCGS. So this will fail. AH
#T one might pick the right ones, but this would be almost the same work.
#T { [1 .. Length(InducedPcgsWrtHomePcgs(cl.centralizer))
#T - Length(InducedPcgsWrtHomePcgs(L))] },
c -> Comm(k, c) in L));
end;
fi;
Info(InfoClasses,1,"Series of sizes ",List(eas,Size));
if mode<3 and InfoLevel(InfoClasses)>1 then
divi:=DivisorsInt(Size(G));
Info(InfoClasses,2,"centsiz: ",divi);
fi;
# Initialize the algorithm for the trivial group.
step:=1;
L :=eas[step];
Lp:=InducedPcgs(home,L);
if not IsIdenticalObj( G, H ) then
Hp := InducedPcgs(home, H );
LcapH := NormalIntersectionPcgs( home, Hp, Lp );
fi;
if mode mod 2=1 # rational classes
or candidates<>false then
mL:=ModuloPcgsByPcSequenceNC(home, home, Lp);
fi;
if mode mod 2=1 then # rational classes
cl:=rec(representative:=One(G),
centralizer:=H,
galoisGroup:=GroupByPrimeResidues([], 1));
cl.galoisGroup!.type:=3;
cl.galoisGroup!.operators:=[];
if mode mod 4=3 then # construct the power tree
cl.power :=rec(representative:=One(G));
cl.power.operator:=One(G);
cl.power.exponent:=1;
cl.power.kernel :=false;
fi;
else
cl:=rec(representative:=One(G),
centralizer:=H);
fi;
if candidates<>false then
cls:=List(candidates, c -> cl);
opr:=List(candidates, c -> One(G));
exp:=ListWithIdenticalEntries(Length(candidates), 1);
else
cls:=[cl];
fi;
# Now go back through the factors by all groups in the elementary abelian
# series.
for step in [step + 1 .. Length(eas)] do
Info(InfoClasses,1,"Step ",step,", ",Length(cls)," classes to lift");
# We apply the homomorphism principle to the homomorphism G/L -> G/K.
# The actual computations are all done in <G>, factors are
# represented by modulo pcgs.
K :=L;
Kp:=Lp;
L :=eas[step];
Lp:=InducedPcgs(home,L);
N :=Kp mod Lp; # modulo pcgs representing the kernel
#T What is this? Obviously it is needed somewhere, but it is
#T certainly not good programming style. AH
SetFilterObj(N, IsPcgs);
if not IsIdenticalObj(G,H) then
KcapH := LcapH;
LcapH := NormalIntersectionPcgs(home,Hp,Lp);
N!.capH:=KcapH mod LcapH;
SetFilterObj( N!.capH, IsPcgs );
else
N!.capH:=N;
fi;
# Rational classes or identification of classes.
if mode mod 2=1
or candidates<>false then
mK:=mL;
mL:=ModuloPcgsByPcSequenceNC(home, home, Lp);
fi;
# Identification of classes.
if candidates<>false then
if mode=4 # test conjugacy of two elements
and not cls[1].representative /
cls[2].representative in K then
return fail;
fi;
blist:=BlistList([1 .. Length(cls)], []);
pos:=Position(blist, false);
while pos<>fail do
# Find a team of candidates with same image under <modK>.
cl:=cls[pos];
cl.representative:=PcElementByExponentsNC(mK,
ExponentsOfPcElement(mK, cl.representative));
cl.candidates:=[];
team:=[];
q:=pos;
while q<>fail do
if cls[q].representative /
cl.representative in K then
c:=candidates[q] ^ opr[q];
if mode mod 2=1 then # rational classes
c:=c ^ exp[q];
fi;
i:=PositionSorted(cl.candidates, c);
if i > Length(cl.candidates)
or cl.candidates[i]<>c then
Add( cl.candidates,c,i);
Add(team, [q], i);
else
Add(team[i], q);
fi;
blist[q]:=true;
fi;
q:=Position(blist, false, q);
od;
# Now <cl> is a class modulo <K> (possibly with
# `<cl>.candidates' a list of elements mapping into this
# class modulo <K>). Let <newcls> be a list of all classes
# modulo <L> that map to <cl> modulo <K> (resp. a list of
# classes to which the list `<cl>.candidates' maps modulo
# <K>, together with `operator's and `exponent's as in
# (c^o^e=r)).
if mode mod 2=1 then # rational classes
newcls:=CentralStepRatClPGroup(home, H, N, mK, mL, cl);
elif cent(cl, N, L) then
newcls:=CentralStepClEANS(home,H, G, N, cl);
else
newcls:=GeneralStepClEANS(home, H, G, N, cl);
fi;
# Update <cls>, <opr> and <exp>.
for i in [1 .. Length(team)] do
for q in team[i] do
cls[q]:=newcls[i];
opr[q]:=opr[q] * newcls[i].operator;
if mode mod 2=1 then # rational classes
ord:=OrderModK(cls[q].representative, mL);
if ord<>1 then
# For historical reasons, the `exponent's
# returns by `CentralStepRatClPGroup' are the
# inverses of what we need.
exp[q]:=exp[q] /
newcls[i].exponent mod ord;
fi;
fi;
od;
od;
pos:=Position(blist, false, pos);
od;
elif mode mod 2=1 then # rational classes
newcls:=[];
for cl in cls do
if IsBound(cl.power) then # construct the power tree
cl.representative:=PcElementByExponentsNC(mK,
ExponentsOfPcElement(mK, cl.representative));
cl.power.representative:=PcElementByExponentsNC(mK,
ExponentsOfPcElement(mK, cl.power.representative));
fi;
new:=CentralStepRatClPGroup(home, G, N, mK, mL, cl);
ord:=OrderModK(new[1].representative, mL);
# if ord <= limit.order
# and ( limit.size=0
# or limit.size mod Size(new[1])=0) then
if IsBound(cl.power) then # construct the power tree
if ord=1 then
power:=cl.power;
else
cl.power.candidates:=[(new[1].representative ^
cl.power.operator) ^ (p*cl.power.exponent)];
power:=CentralStepRatClPGroup(home, G, N, mK, mL,
cl.power)[1];
power.operator:=cl.power.operator
* power.operator;
power.exponent:=cl.power.exponent
/ power.exponent mod ord;
fi;
for c in new do
c.power:=power;
od;
fi;
Append(newcls, new);
# fi
od;
cls:=newcls;
else
newcls:=[];
for cl in cls do
#if consider=true or consider(fhome,cl.representative,cl.centralizerpcgs,K,L)
#then
if cent(cl, N, L) then
news:=CentralStepClEANS(home,G, G, N, cl);
else
news:=GeneralStepClEANS(home, G, G, N, cl);
fi;
Assert(1,ForAll(news,
i->ForAll(GeneratorsOfGroup(i.centralizer),
j->Comm(i.representative,j) in eas[step])));
Append(newcls,news);
#fi;
od;
cls:=newcls;
fi;
if InfoLevel(InfoClasses)>1 then
c:=Collected(List(cls,i->Size(i.centralizer)));
if not IsBound( divi ) then
divi:=DivisorsInt(Size(G));
fi;
c:=Concatenation(c,List(divi,i->[i,0])); # to cope with `First'
Info(InfoClasses,2,List(divi,i->First(c,j->j[1]=i)[2]));
fi;
od;
if mode=4 then # test conjugacy of two elements
if cls[1].representative<>cls[2].representative then
return fail;
else
return opr[1] / opr[2];
fi;
fi;
if candidates<>false then # add operators (and exponents)
for i in [1 .. Length(cls)] do
cls[i].operator:=opr[i];
if mode mod 2=1 then # rational classes
cls[i].exponent:=exp[i];
fi;
od;
fi;
return cls;
end);
#############################################################################
##
#F OrderModK( <h>, <mK> ) . . . . . . . . . . order modulo normal subgroup
##
InstallGlobalFunction( OrderModK, function( h, mK )
local ord, d, o;
ord:=1;
d:=DepthOfPcElement( mK, h );
while d <= Length( mK ) do
o:=RelativeOrders( mK )[ d ];
h:=h ^ o;
ord:=ord * o;
d:=DepthOfPcElement( mK, h, d + 1 );
od;
return ord;
end );
#############################################################################
##
#F OldSubspaceVectorSpaceGroup( <N>, <p>, <gens>, <howmuch> ) . complement and projection
##
## This function creates a record containing information about a complement
## in <N> to the span of <gens>.
##
BindGlobal("OldSubspaceVectorSpaceGroup", function( N, p, gens )
local zero, one, r, ran, n, nan, cg, pos, Q, i, j, v;
one:=One( GF( p ) ); zero:=0 * one;
r:=Length( N ); ran:=[ 1 .. r ];
n:=Length( gens ); nan:=[ 1 .. n ];
Q:=[ ];
if n <> 0 and IsMultiplicativeElementWithInverse( gens[ 1 ] ) then
Q:=List( gens, gen -> ExponentsOfPcElement( N, gen ) ) * one;
else
Q:=ShallowCopy( gens );
fi;
cg:=rec( matrix :=[ ],
needed := [],
one :=one,
baseComplement:=ShallowCopy( ran ),
projection := IdentityMat( r, one ),
commutator :=0,
centralizer :=0,
dimensionN :=r,
dimensionC :=n );
if n = 0 or r = 0 then
cg.inverse:=NullMapMatrix;
return cg;
fi;
for i in nan do
cg.matrix[ i ]:=Concatenation( Q[ i ], zero * nan );
cg.matrix[ i ][ r + i ]:=one;
od;
TriangulizeMat( cg.matrix );
pos:=1;
for v in cg.matrix do
while v[ pos ] = zero do
pos:=pos + 1;
od;
RemoveSet( cg.baseComplement, pos );
if pos <= r then cg.commutator :=cg.commutator + 1;
else cg.centralizer:=cg.centralizer + 1; fi;
od;
cg.needed :=[ ];
cg.projection :=IdentityMat( r, one );
# Find a right pseudo inverse for <Q>.
Append( Q, cg.projection );
Q:=MutableTransposedMat( Q );
TriangulizeMat( Q );
Q:=TransposedMat( Q );
i:=1;
j:=1;
while i <= Length( N ) do
while j <= Length( gens ) and Q[ j ][ i ] = zero do
j:=j + 1;
od;
if j <= Length( gens ) and Q[ j ][ i ] <> zero then
cg.needed[ i ]:=j;
else
# If <Q> does not have full rank, terminate when the bottom row
# is reached.
i:=Length( N );
fi;
i:=i + 1;
od;
if IsEmpty( cg.needed ) then
cg.inverse:=NullMapMatrix;
else
cg.inverse:=Q{ Length( gens ) + ran }
{ [ 1 .. Length( cg.needed ) ] };
cg.inverse:=ImmutableMatrix(p,cg.inverse,true);
fi;
if IsEmpty( cg.baseComplement ) then
cg.projection:=NullMapMatrix;
else
# Find a base change matrix for the projection onto the complement.
for i in [ 1 .. cg.commutator ] do
cg.projection[ i ][ i ]:=zero;
od;
Q:=[ ];
for i in [ 1 .. cg.commutator ] do
Q[ i ]:=cg.matrix[ i ]{ ran };
od;
for i in [ cg.commutator + 1 .. r ] do
Q[ i ]:=ListWithIdenticalEntries( r, zero );
Q[ i ][ cg.baseComplement[ i-r+Length(cg.baseComplement) ] ]
:=one;
od;
cg.projection:=cg.projection ^ Q;
cg.projection:=cg.projection{ ran }{ cg.baseComplement };
cg.projection:=ImmutableMatrix(p,cg.projection,true);
fi;
return Immutable(cg);
end );
#############################################################################
##
#F OldKernelHcommaC( <N>, <h>, <C> )
##
## Given a homomorphism C -> N, c |-> [h,c], this function determines (a) a
## vector space decomposition N = [h,C] + K with projection onto K and (b)
## the ``kernel'' S < C which plays the role of C_G(h) in lemma 3.1 of
## [Mecky, Neub\"user, Bull. Aust. Math. Soc. 40].
##
BindGlobal("OldKernelHcommaC", function( N, h, C )
local i, tmp, v;
N!.subspace := OldSubspaceVectorSpaceGroup( N, RelativeOrders( N )[ 1 ],
List( C, c -> Comm( h, c ) ) );
tmp := [ ];
for i in [ N!.subspace.commutator + 1 ..
N!.subspace.commutator + N!.subspace.centralizer ] do
v := N!.subspace.matrix[ i ];
tmp[ i - N!.subspace.commutator ] := PcElementByExponentsNC( C,
v{ [ N!.subspace.dimensionN + 1 ..
N!.subspace.dimensionN + N!.subspace.dimensionC ] } );
od;
return tmp;
end );
#############################################################################
##
#F CentralStepConjugatingElement( ... ) . . . . . . . . . . . . . . . local
##
## This function returns an element of <G> conjugating <hk1> to <hk2>^<l>.
##
InstallGlobalFunction( CentralStepConjugatingElement,
function( N, h, k1, k2, l, cN )
local v, conj;
v:=ExponentsOfPcElement( N, h ^ -l * h ^ cN * k1 * k2 ^ -l );
conj:=LinearCombinationPcgs( N!.CmodK{ N!.subspace.needed },
v * N!.subspace.inverse,OneOfPcgs( N ) );
conj:=LeftQuotient( conj, cN );
return conj;
end );
#############################################################################
##
#F CentralStepRatClPGroup(<homepcgs>, <G>, <N>, <mK>, <mL>, <cl> )
##
InstallGlobalFunction( CentralStepRatClPGroup,
function( home, G, N, mK, mL, cl )
local h, # preimage of `cl.representative' under <hom>
candexps, # list of exponent vectors for <h> mod <candidates>
classes, # the resulting list of classes
ohN, oh, # order of <h> in `Range(<hom>)' resp. `Source(<hom>)'
p, # exponent of <N>
K, # a complement to $[h,C]$ in <N>
Gal, gal, # Galois group for element in `Source(<hom>)'
preimage, # preimage of $Gal(hN)$ in $Z_oh^*$
operator, # generator of <preimage> acting by conjugation
reps, conj, #\ representatives, conjugating elements,
exps, #/ exponents and orbit lengths in orbit algorithm
Q, v, r, # subspace to be projected onto, projection vectors
k, # orbit representative in <N>
gens, oprs, # generators and operators for new Galois group
type, # the type of the Galois group as subgroup of Z_2^r^*
i, j, l, c, # loop variables
C, cyc, xset, opr, orb,kern,img;
p :=RelativeOrders( N )[ 1 ];
h :=cl.representative;
ohN:=OrderModK( h, mK );
oh :=OrderModK( h, mL );
classes:=[ ];
if oh = 1 then
# Special case: <h> is trivial.
Gal:=Units( Integers mod 1 );
gal:=GroupByPrimeResidues( [ ], p );
gal!.type:=3;
gal!.operators:=[ ];
if IsBound( cl.candidates ) then
for c in cl.candidates do
l:=LeadingExponentOfPcElement( N, c );
if l = fail then
l:=1;
c:=rec( representative:=c,
galoisGroup:=TrivialSubgroup( Gal ) );
c.galoisGroup!.type:=3;
c.galoisGroup!.operators:=[ ];
else
c:=rec( representative:=c ^ ( 1 / l mod p ),
galoisGroup:=gal );
fi;
c.centralizer:=G;
c.operator :=OneOfPcgs( N );
c.exponent :=l;
Add( classes, c );
od;
else
c:=rec( representative:=One( G ),
centralizer:=G,
galoisGroup:=TrivialSubgroup( Gal ) );
c.galoisGroup!.type:=3;
c.galoisGroup!.operators:=[ ];
Add( classes, c );
for v in EnumeratorOfNormedRowVectors( GF( p ) ^ Length( N ) ) do
c:=rec( representative:=PcElementByExponentsNC( N, v ),
centralizer:=G,
galoisGroup:=gal );
Add( classes, c );
od;
fi;
else
Gal:=Units( Integers mod oh );
if IsBound( cl.kernel ) then
N:=cl.kernel;
else
N!.CmodK:=InducedPcgs(home, cl.centralizer ) mod
DenominatorOfModuloPcgs( N );
kern:=DenominatorOfModuloPcgs( N );
img:=OldKernelHcommaC( N, h, N!.CmodK ) ;
#N!.CmodL:=ExtendedPcgs(kern,img);
N!.CmodL:=InducedPcgsByPcSequenceAndGenerators(ParentPcgs( kern ),
kern, img );
fi;
if IsBound( cl.candidates ) then
cl.candidates:=List( cl.candidates, c ->
LeftQuotient( h, c ) );
candexps:=List( cl.candidates, c ->
ExponentsOfPcElement( N, c ) ) * N!.subspace.projection;
fi;
# If <p> = 2, use a projection operation.
if p = 2 then
# Construct the preimage of $Gal(hN)$ in $Z_oh^*$.
if ohN <= 2 then
preimage:=GroupByPrimeResidues( [ -1, 5 ], oh );
preimage!.type:=1;
preimage!.operators:=List( GeneratorsOfGroup( preimage ),
i -> One( G ) );
else
if cl.galoisGroup!.type = 1 then
preimage:=[ -1, 5^(ohN/(2*Size(cl.galoisGroup))) ];
elif cl.galoisGroup!.type = 2 then
preimage:=[ -( 5^(ohN/(4*Size(cl.galoisGroup)))) ];
else
preimage:=[ 5^(ohN/(4*Size(cl.galoisGroup))) ];
fi;
preimage:=GroupByPrimeResidues( preimage, oh );
preimage!.type:=cl.galoisGroup!.type;
if Length( GeneratorsOfGroup( preimage ) ) =
Length( GeneratorsOfGroup( cl.galoisGroup ) ) then
preimage!.operators:=cl.galoisGroup!.operators;
else
preimage!.operators:=Concatenation
( cl.galoisGroup!.operators, [ One( G ) ] );
fi;
fi;
# Construct the image of the homomorphism <preimage> -> <K>.
Q:=[ ];
for i in [ 1 .. Length( GeneratorsOfGroup( preimage ) ) ] do
#Assert(2,LeftQuotient(h^Int(GeneratorsOfGroup(preimage)[i]),
# h^preimage!.operators[i]) in
# Group(NumeratorOfModuloPcgs(N)));
Add( Q, ExponentsOfPcElement( N, LeftQuotient( h ^
Int( GeneratorsOfGroup( preimage )[ i ] ),
h ^ preimage!.operators[ i ] ) ) );
od;
Q:=Q * N!.subspace.projection;
K:=InducedPcgsByPcSequenceNC( N,
N{ N!.subspace.baseComplement } );
K!.subspace:=OldSubspaceVectorSpaceGroup( K, p, Q );
# Project the factors in <N> onto a complement to <Q>.
if IsBound( cl.candidates ) then
v:=List( candexps, ShallowCopy );
r:=v * K!.subspace.projection;
reps:=[ ];
exps:=[ ];
conj:=[ ];
if not IsEmpty( K!.subspace.baseComplement ) then
v{[1..Length(v)]}{K!.subspace.baseComplement}:=
v{[1..Length(v)]}{K!.subspace.baseComplement} + r;
fi;
v:=v * K!.subspace.inverse;
for i in [ 1 .. Length( r ) ] do
reps[ i ]:=PcElementByExponentsNC
( K, K!.subspace.baseComplement, r[ i ] );
exps[ i ]:=LinearCombinationPcgs(
GeneratorsOfGroup(preimage){K!.subspace.needed},
v[ i ],One(preimage));
conj[ i ]:=LinearCombinationPcgs(
preimage!.operators { K!.subspace.needed }, v[ i ],
One(G));
od;
# In the construction case, the complement to <Q> is a set of
# representatives.
else
reps:=EnumeratorByPcgs( K, K!.subspace.baseComplement );
fi;
# The kernel of the homomorphism into <K> is the Galois group of
# <h>.
if IsTrivial( preimage ) then # pre = < 1 >
gens:=GeneratorsOfGroup( preimage );
oprs:=preimage!.operators;
type:=preimage!.type;
else
if Q[ 1 ] = Zero( Q[ 1 ] ) then i:=1;
else i:=2; fi;
if Length( GeneratorsOfGroup( preimage ) ) = 1 then
gens:=[ GeneratorsOfGroup( preimage )[ 1 ] ^ i ];
oprs:=[ preimage!.operators [ 1 ] ^ i ];
if preimage!.type = 1 then type:=2 * i - 1; # <-1>
elif preimage!.type = 2 then type:=i + 1;
else type:=3; fi;
else
if Q[ 2 ] = Zero( Q[ 2 ] ) then j:=1;
else j:=2; fi;
if i = 1 then
gens:=[ GeneratorsOfGroup( preimage )[ 1 ],
GeneratorsOfGroup( preimage )[ 2 ] ^ j ];
oprs:=[ preimage!.operators [ 1 ],
preimage!.operators [ 2 ] ^ j ];
type:=1;
elif j = 2 and Q[ 1 ] = Q[ 2 ] then
gens:=[ GeneratorsOfGroup( preimage )[ 1 ] *
GeneratorsOfGroup( preimage )[ 2 ] ];
oprs:=[ preimage!.operators [ 1 ] *
preimage!.operators [ 2 ] ];
type:=2;
else
gens:=[ GeneratorsOfGroup( preimage )[ 2 ] ^ j ];
oprs:=[ preimage!.operators [ 2 ] ^ j ];
type:=3;
fi;
fi;
fi;
# If <p> <> 2, use an affine operation of a cyclic group generated by
# <preimage>.
else
K:=EnumeratorByPcgs( N, N!.subspace.baseComplement );
cyc:=GroupByPrimeResidues( [ PowerModInt
( PrimitiveRootMod( oh ),
IndexInParent( cl.galoisGroup ), oh ) ], oh );
SetSize( cyc, Phi( oh ) / IndexInParent( cl.galoisGroup ) );
if IsTrivial( cyc ) then
preimage:=One( cyc );
else
SetIndependentGeneratorsOfAbelianGroup( cyc,
GeneratorsOfGroup( cyc ) );
preimage:=Pcgs( cyc )[ 1 ];
fi;
if IsTrivial( cl.galoisGroup ) then
operator:=One( G );
else
operator:=cl.galoisGroup!.operators[ 1 ];
fi;
v:=PcElementByExponentsNC( N, N!.subspace.baseComplement,
ExponentsOfPcElement( N, LeftQuotient( h ^ Int( preimage ),
h ^ operator ) ) * N!.subspace.projection );
opr:=function( k, l )
return
#AH, jun3 2001: without the pcgs filtereing we might get
# extra kernel elements. I have no idea how this was
# originally avoided. This is rather a workaround than a fix
# -- the whole code should be rewritten cleanly.
PcElementByExponentsNC(N,ExponentsOfPcElement(N,
( v * k ) ^ ( 1 / Int( l ) mod p )
));
end;
xset:=ExternalSet( cyc, K, opr );
reps:=[ ];
exps:=[ ];
if IsBound( cl.candidates ) then
conj:=[ ];
for c in candexps do
orb:=ExternalOrbit( xset, PcElementByExponentsNC( N,
N!.subspace.baseComplement, c ) );
Add( reps, CanonicalRepresentativeOfExternalSet( orb ) );
i:=Size( cyc ) / Order( ActorOfExternalSet( orb ) );
Add( exps, preimage ^ i );
Add( conj, operator ^ i );
od;
else
for orb in ExternalOrbits( xset ) do
Add( reps, CanonicalRepresentativeOfExternalSet( orb ) );
Add( exps, preimage ^ Size( orb ) );
od;
fi;
fi;
# If <reps> is a set of representatives of the orbits then <h><reps>
# is a set of representatives of the rational classes in <hN>.
for l in [ 1 .. Length( reps ) ] do
k:=reps[ l ];
# Construct the Galois group and find conjugating elements
# corresponding to its generator(s).
if p <> 2 then
gens:=[ exps[ l ] ];
oprs:=[ operator ^ Int( exps[ l ] ) ];
fi;
gal:=SubgroupNC( Gal, gens );
if p = 2 then
gal!.type:=type;
fi;
gal!.operators:=[ ];
for i in [ 1 .. Length( GeneratorsOfGroup( gal ) ) ] do
Add( gal!.operators, CentralStepConjugatingElement
( N, h, k, k, Int( GeneratorsOfGroup( gal )[ i ] ),
oprs[ i ] ) );
od;
C:=SubgroupNC( G, N!.CmodL );
c:=rec( representative:=h * k,
centralizer:=C,
galoisGroup:=gal );
if IsBound( cl.candidates ) then
# cl.candidates[l] ^ c.operator =
# c.representative ^ c.exponent (DIFFERS from (c^o^e=r)!)
c.exponent:=Int( exps[ l ] );
c.operator:=CentralStepConjugatingElement
( N, h, cl.candidates[ l ], k, c.exponent, conj[ l ] );
if IsBound( cl.kernel ) then
c.kernel:=N;
fi;
fi;
Add( classes, c );
od;
fi;
return classes;
end );
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