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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Alexander Hulpke.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the declarations of operations for factor group maps
##
#############################################################################
##
#M NaturalHomomorphismsPool(G) . . . . . . . . . . . . . . initialize method
##
InstallMethod(NaturalHomomorphismsPool,true,[IsGroup],0,
G->rec(GopDone:=false,ker:=[],ops:=[],cost:=[],group:=G,lock:=[],
intersects:=[],blocksdone:=[],in_code:=false,dotriv:=false));
#############################################################################
##
#F EraseNaturalHomomorphismsPool(G) . . . . . . . . . . . . initialize
##
InstallGlobalFunction(EraseNaturalHomomorphismsPool,function(G)
local r;
r:=NaturalHomomorphismsPool(G);
if r.in_code=true then return;fi;
r.GopDone:=false;
r.ker:=[];
r.ops:=[];
r.cost:=[];
r.group:=G;
r.lock:=[];
r.intersects:=[];
r.blocksdone:=[];
r.in_code:=false;
r.dotriv:=false;
r:=NaturalHomomorphismsPool(G);
end);
#############################################################################
##
#F AddNaturalHomomorphismsPool(G,N,op[,cost[,blocksdone]]) . Store operation
## op for kernel N if there is not already a cheaper one
## returns false if nothing had been added and 'fail' if adding was
## forbidden
##
InstallGlobalFunction(AddNaturalHomomorphismsPool,function(arg)
local G, N, op, pool, p, c, perm, ch, diff, nch, nd, involved, i;
G:=arg[1];
N:=arg[2];
op:=arg[3];
# don't store trivial cases
if Size(N)=Size(G) then
Info(InfoFactor,4,"full group");
return false;
elif Size(N)=1 then
# do we really want the trivial subgroup?
if not (HasNaturalHomomorphismsPool(G) and
NaturalHomomorphismsPool(G).dotriv=true) then
Info(InfoFactor,4,"trivial sub: ignore");
return false;
fi;
Info(InfoFactor,4,"trivial sub: OK");
fi;
pool:=NaturalHomomorphismsPool(G);
# split lists in their components
if IsList(op) and not IsInt(op[1]) then
p:=[];
for i in op do
if IsMapping(i) then
c:=Intersection(G,KernelOfMultiplicativeGeneralMapping(i));
else
c:=Core(G,i);
fi;
Add(p,c);
AddNaturalHomomorphismsPool(G,c,i);
od;
# transfer in numbers list
op:=List(p,i->PositionSet(pool.ker,i));
if Length(arg)<4 then
# add the prices
c:=Sum(pool.cost{op});
fi;
# compute/get costs
elif Length(arg)>3 then
c:=arg[4];
else
if IsGroup(op) then
c:=IndexNC(G,op);
elif IsMapping(op) then
c:=Image(op);
if IsPcGroup(c) then
c:=1;
elif IsPermGroup(c) then
c:=NrMovedPoints(c);
else
c:=Size(c);
fi;
fi;
fi;
# check whether we have already a better operation (or whether this normal
# subgroup is locked)
p:=PositionSet(pool.ker,N);
if p=fail then
if pool.in_code then
return fail;
fi;
p:=PositionSorted(pool.ker,N);
# compute the permutation we have to apply finally
perm:=PermList(Concatenation([1..p-1],[Length(pool.ker)+1],
[p..Length(pool.ker)]))^-1;
# first add at the end
p:=Length(pool.ker)+1;
pool.ker[p]:=N;
Info(InfoFactor,2,"Added price ",c," for size ",IndexNC(G,N),
" in group of size ",Size(G));
elif c>=pool.cost[p] then
Info(InfoFactor,4,"bad price");
return false; # nothing added
elif pool.lock[p]=true then
return fail; # nothing added
else
Info(InfoFactor,2,"Changed price ",c," for size ",IndexNC(G,N));
perm:=();
# update dependent costs
ch:=[p];
diff:=[pool.cost[p]-c];
while Length(ch)>0 do
nch:=[];
nd:=[];
for i in [1..Length(pool.ops)] do
if IsList(pool.ops[i]) then
involved:=Intersection(pool.ops[i],ch);
if Length(involved)>0 then
involved:=Sum(diff{List(involved,x->Position(ch,x))});
pool.cost[i]:=pool.cost[i]-involved;
Add(nch,i);
Add(nd,involved);
fi;
fi;
od;
ch:=nch;
diff:=nd;
od;
fi;
if IsMapping(op) and not HasKernelOfMultiplicativeGeneralMapping(op) then
SetKernelOfMultiplicativeGeneralMapping(op,N);
fi;
pool.ops[p]:=op;
pool.cost[p]:=c;
pool.lock[p]:=false;
# update the costs of all intersections that are affected
for i in [1..Length(pool.ker)] do
if IsList(pool.ops[i]) and IsInt(pool.ops[i][1]) and p in pool.ops[i] then
pool.cost[i]:=Sum(pool.cost{pool.ops[i]});
fi;
od;
if Length(arg)>4 then
pool.blocksdone[p]:=arg[5];
else
pool.blocksdone[p]:=false;
fi;
if perm<>() then
# sort the kernels anew
pool.ker:=Permuted(pool.ker,perm);
# sort/modify the other components accordingly
pool.ops:=Permuted(pool.ops,perm);
for i in [1..Length(pool.ops)] do
# if entries are lists of integers
if IsList(pool.ops[i]) and IsInt(pool.ops[i][1]) then
pool.ops[i]:=List(pool.ops[i],i->i^perm);
fi;
od;
pool.cost:=Permuted(pool.cost,perm);
pool.lock:=Permuted(pool.lock,perm);
pool.blocksdone:=Permuted(pool.blocksdone,perm);
pool.intersects:=Set(pool.intersects,i->List(i,j->j^perm));
fi;
return perm; # if anyone wants to keep the permutation
end);
#############################################################################
##
#F LockNaturalHomomorphismsPool(G,N) . . store flag to prohibit changes of
## the map to N
##
InstallGlobalFunction(LockNaturalHomomorphismsPool,function(G,N)
local pool;
pool:=NaturalHomomorphismsPool(G);
N:=PositionSet(pool.ker,N);
if N<>fail then
pool.lock[N]:=true;
fi;
end);
#############################################################################
##
#F UnlockNaturalHomomorphismsPool(G,N) . . . clear flag to allow changes of
## the map to N
##
InstallGlobalFunction(UnlockNaturalHomomorphismsPool,function(G,N)
local pool;
pool:=NaturalHomomorphismsPool(G);
N:=PositionSet(pool.ker,N);
if N<>fail then
pool.lock[N]:=false;
fi;
end);
#############################################################################
##
#F KnownNaturalHomomorphismsPool(G,N) . . . . . check whether Hom is stored
## (or obvious)
##
InstallGlobalFunction(KnownNaturalHomomorphismsPool,function(G,N)
return N=G or Size(N)=1
or PositionSet(NaturalHomomorphismsPool(G).ker,N)<>fail;
end);
#############################################################################
##
#F GetNaturalHomomorphismsPool(G,N) . . . . get operation for G/N if known
##
InstallGlobalFunction(GetNaturalHomomorphismsPool,function(G,N)
local pool,p,h,ise,emb,i,j;
if not HasNaturalHomomorphismsPool(G) then
return fail;
fi;
pool:=NaturalHomomorphismsPool(G);
p:=PositionSet(pool.ker,N);
if p<>fail then
h:=pool.ops[p];
if IsList(h) then
# just stored as intersection. Construct the mapping!
# join intersections
ise:=ShallowCopy(h);
for i in ise do
if IsList(pool.ops[i]) and IsInt(pool.ops[i][1]) then
for j in Filtered(pool.ops[i],j-> not j in ise) do
Add(ise,j);
od;
elif not pool.blocksdone[i] then
h:=GetNaturalHomomorphismsPool(G,pool.ker[i]);
pool.in_code:=true; # don't add any new kernel here
# (which would mess up the numbering)
ImproveActionDegreeByBlocks(G,pool.ker[i],h);
pool.in_code:=false;
fi;
od;
ise:=List(ise,i->GetNaturalHomomorphismsPool(G,pool.ker[i]));
if not (ForAll(ise,IsPcGroup) or ForAll(ise,IsPermGroup)) then
ise:=List(ise,x->x*IsomorphismPermGroup(Image(x)));
fi;
h:=CallFuncList(DirectProduct,List(ise,Image));
emb:=List([1..Length(ise)],i->Embedding(h,i));
emb:=List(GeneratorsOfGroup(G),
i->Product([1..Length(ise)],j->Image(emb[j],Image(ise[j],i))));
ise:=SubgroupNC(h,emb);
h:=GroupHomomorphismByImagesNC(G,ise,GeneratorsOfGroup(G),emb);
SetKernelOfMultiplicativeGeneralMapping(h,N);
pool.ops[p]:=h;
elif IsGroup(h) then
h:=FactorCosetAction(G,h,N); # will implicitly store
fi;
p:=h;
fi;
return p;
end);
#############################################################################
##
#F DegreeNaturalHomomorphismsPool(G,N) degree for operation for G/N if known
##
InstallGlobalFunction(DegreeNaturalHomomorphismsPool,function(G,N)
local p,pool;
pool:=NaturalHomomorphismsPool(G);
p:=First([1..Length(pool.ker)],i->IsIdenticalObj(pool.ker[i],N));
if p=fail then
p:=PositionSet(pool.ker,N);
fi;
if p<>fail then
p:=pool.cost[p];
fi;
return p;
end);
#############################################################################
##
#F CloseNaturalHomomorphismsPool(<G>[,<N>]) . . calc intersections of known
## operation kernels, don't continue anything which is smaller than N
##
InstallGlobalFunction(CloseNaturalHomomorphismsPool,function(arg)
local G,pool,p,comb,i,c,perm,isi,N,discard,Npos,psub,pder,new,co,pos,j,k;
G:=arg[1];
pool:=NaturalHomomorphismsPool(G);
p:=[1..Length(pool.ker)];
Npos:=fail;
if Length(arg)>1 then
# get those p that lie above N
N:=arg[2];
p:=Filtered(p,i->IsSubset(pool.ker[i],N));
if Length(p)=0 then
return;
fi;
SortParallel(List(pool.ker{p},Size),p);
if Size(pool.ker[p[1]])=Size(N) then
# N in pool
Npos:=p[1];
c:=pool.cost[Npos];
p:=Filtered(p,x->pool.cost[x]<c);
fi;
else
SortParallel(List(pool.ker{p},Size),p);
N:=fail;
fi;
if Size(Intersection(pool.ker{p}))>Size(N) then
# cannot reach N
return;
fi;
# determine inclusion, derived
psub:=List(pool.ker,x->0);
pder:=List(pool.ker,x->0);
discard:=[];
for i in [1..Length(p)] do
c:=Filtered(p{[1..i-1]},x->IsSubset(pool.ker[p[i]],pool.ker[x]));
psub[p[i]]:=Set(c);
if ForAny(c,x->pool.cost[x]<=pool.cost[p[i]]) then
AddSet(discard,p[i]);
fi;
c:=DerivedSubgroup(pool.ker[p[i]]);
if N<>fail then c:=ClosureGroup(N,c);fi;
pder[p[i]]:=Position(pool.ker,c);
od;
#if Length(discard)>0 then Error(discard);fi;
#discard:=[];
p:=Filtered(p,x->not x in discard);
for i in discard do psub[i]:=0;od;
new:=p;
repeat
# now intersect, staring from top
if new=p then
comb:=Combinations(new,2);
else
comb:=List(Cartesian(p,new),Set);
fi;
comb:=Filtered(comb,i->not i in pool.intersects and Length(i)>1);
Info(InfoFactor,2,"CloseNaturalHomomorphismsPool: ",Length(comb));
new:=[];
discard:=[];
i:=1;
while i<=Length(comb) do
co:=comb[i];
# unless they contained in each other
if not (co[1] in psub[co[2]] or co[2] in psub[co[1]]
# or there a subgroup below both that is already at least as cheap
or ForAny(Intersection(psub[co[1]],psub[co[2]]),
x->pool.cost[x]<=pool.cost[co[1]]+pool.cost[co[2]])
# or both intersect in an abelian factor?
or (N<>fail and pder[co[1]]<>fail and pder[co[1]]=pder[co[2]]
and pder[co[1]]<>Npos)) then
c:=Intersection(pool.ker[co[1]],pool.ker[co[2]]);
pos:=Position(pool.ker,c);
if pos=fail or pool.cost[pos]>pool.cost[co[1]]+pool.cost[co[2]] then
Info(InfoFactor,3,"Intersect ",co,": ",
Size(pool.ker[co[1]])," ",Size(pool.ker[co[2]]),
" yields ",Size(c));
isi:=ShallowCopy(co);
# unpack 'iterated' lists
if IsList(pool.ops[co[2]]) and IsInt(pool.ops[co[2]][1]) then
isi:=Concatenation(isi{[1]},pool.ops[co[2]]);
fi;
if IsList(pool.ops[co[1]]) and IsInt(pool.ops[co[1]][1]) then
isi:=Concatenation(isi{[2..Length(isi)]},pool.ops[co[1]]);
fi;
isi:=Set(isi);
perm:=AddNaturalHomomorphismsPool(G,c,isi,Sum(pool.cost{co}));
if pos=fail then
pos:=Position(pool.ker,c);
p:=List(p,i->i^perm);
new:=List(new,i->i^perm);
discard:=OnSets(discard,perm);
#pder:=Permuted(List(pder,x->x^perm),perm);
for k in [1..Length(pder)] do
if IsPosInt(pder[k]) then pder[k]:=pder[k]^perm;fi;
od;
Add(pder,0);
pder:=Permuted(pder,perm);
#psub:=Permuted(List(psub,x->OnTuples(x,perm)));
for k in [1..Length(psub)] do
if IsList(psub[k]) then psub[k]:=OnSets(psub[k],perm);fi;
od;
Add(psub,0);
psub:=Permuted(psub,perm);
Apply(comb,j->OnSets(j,perm));
# add new c if needed
for j in p do
if IsSubset(pool.ker[j],c) then
AddSet(psub[j],pos);
if pool.cost[j]>=pool.cost[pos] then
AddSet(discard,j);
fi;
fi;
od;
psub[pos]:=Set(Filtered(p,x->IsSubset(c,pool.ker[x])));
pder[pos]:=fail;
else
if perm<>() then Error("why perm here?");fi;
psub[pos]:=Set(Filtered(p,x->IsSubset(c,pool.ker[x])));
fi;
AddSet(new,pos);
if ForAny(psub[pos],x->pool.cost[x]<=pool.cost[pos]) then
AddSet(discard,pos);
fi;
pder[pos]:=fail;
if c=N and pool.cost[pos]^3<=IndexNC(G,N) then
return; # we found something plausible
fi;
else
Info(InfoFactor,5,"Intersect ",co,": ",
Size(pool.ker[co[1]])," ",Size(pool.ker[co[2]]),
" yields ",Size(c));
fi;
fi;
i:=i+1;
od;
#discard:=[];
for i in discard do psub[i]:=0;od;
p:=Difference(Union(p,new),discard);
new:=Difference(new,discard);
SortParallel(List(pool.ker{p},Size),p);
SortParallel(List(pool.ker{new},Size),new);
until Length(new)=0;
end);
#############################################################################
##
#F FactorCosetAction( <G>, <U>, [<N>] ) operation on the right cosets Ug
## with possibility to indicate kernel
##
InstallGlobalFunction("DoFactorCosetAction",function(arg)
local G,u,op,h,N,rt;
G:=arg[1];
u:=arg[2];
if Length(arg)>2 then
N:=arg[3];
else
N:=false;
fi;
if IsList(u) and Length(u)=0 then
u:=G;
Error("only trivial operation ? I Set u:=G;");
fi;
if N=false then
N:=Core(G,u);
fi;
rt:=RightTransversal(G,u);
if not IsRightTransversalRep(rt) then
# the right transversal has no special `PositionCanonical' method.
rt:=List(rt,i->RightCoset(u,i));
fi;
h:=ActionHomomorphism(G,rt,OnRight,"surjective");
op:=Image(h,G);
SetSize(op,IndexNC(G,N));
# and note our knowledge
SetKernelOfMultiplicativeGeneralMapping(h,N);
AddNaturalHomomorphismsPool(G,N,h);
return h;
end);
InstallMethod(FactorCosetAction,"by right transversal operation",
IsIdenticalObj,[IsGroup,IsGroup],0,
function(G,U)
return DoFactorCosetAction(G,U);
end);
InstallOtherMethod(FactorCosetAction,
"by right transversal operation, given kernel",IsFamFamFam,
[IsGroup,IsGroup,IsGroup],0,
function(G,U,N)
return DoFactorCosetAction(G,U,N);
end);
InstallMethod(FactorCosetAction,"by right transversal operation, Niceo",
IsIdenticalObj,[IsGroup and IsHandledByNiceMonomorphism,IsGroup],0,
function(G,U)
local hom;
hom:=RestrictedNiceMonomorphism(NiceMonomorphism(G),G);
return hom*DoFactorCosetAction(NiceObject(G),Image(hom,U));
end);
InstallOtherMethod(FactorCosetAction,
"by right transversal operation, given kernel, Niceo",IsFamFamFam,
[IsGroup and IsHandledByNiceMonomorphism,IsGroup,IsGroup],0,
function(G,U,N)
local hom;
hom:=RestrictedNiceMonomorphism(NiceMonomorphism(G),G);
return hom*DoFactorCosetAction(NiceObject(G),Image(hom,U),Image(hom,N));
end);
# action on lists of subgroups
InstallOtherMethod(FactorCosetAction,
"On cosets of list of groups",IsElmsColls,
[IsGroup,IsList],0,
function(G,L)
local q,i,gens,imgs,d;
if Length(L)=0 or not ForAll(L,x->IsGroup(x) and IsSubset(G,x)) then
TryNextMethod();
fi;
q:=List(L,x->FactorCosetAction(G,x));
gens:=MappingGeneratorsImages(q[1])[1];
imgs:=List(q,x->List(gens,y->ImagesRepresentative(x,y)));
d:=imgs[1];
for i in [2..Length(imgs)] do
d:=SubdirectDiagonalPerms(d,imgs[i]);
od;
imgs:=Group(d);
q:=GroupHomomorphismByImagesNC(G,imgs,gens,d);
return q;
end);
#############################################################################
##
#M DoCheapActionImages(G) . . . . . . . . . . All cheap operations for G
##
InstallMethod(DoCheapActionImages,"generic",true,[IsGroup],0,Ignore);
InstallMethod(DoCheapActionImages,"permutation",true,[IsPermGroup],0,
function(G)
local pool, dom, o, op, Go, j, b, i,allb,newb,mov,allbold,onlykernel,k,
found,type;
onlykernel:=ValueOption("onlykernel");
found:=NrMovedPoints(G);
pool:=NaturalHomomorphismsPool(G);
if pool.GopDone=false then
dom:=MovedPoints(G);
# orbits
o:=OrbitsDomain(G,dom);
o:=Set(o,Set);
# do orbits and test for blocks
for i in o do
if Length(i)<>Length(dom) or
# only works if domain are the first n points
not (1 in dom and 2 in dom and IsRange(dom)) then
op:=ActionHomomorphism(G,i,"surjective");
Range(op:onlyimage); #`onlyimage' forces same generators
AddNaturalHomomorphismsPool(G,Stabilizer(G,i,OnTuples),
op,Length(i));
type:=1;
else
op:=IdentityMapping(G);
type:=2;
fi;
Go:=Image(op,G);
# all minimal and maximal blocks
mov:=MovedPoints(Go);
allb:=ShallowCopy(RepresentativesMinimalBlocks(Go,mov));
allbold:=[];
SortBy(allb,Length);
while Length(allb)>0 do
j:=Remove(allb);
Add(allbold,j);
# even if generic spread, <found, since blocks are always of size
# >1.
if Length(i)/Length(j)<found then
b:=List(Orbit(G,i{j},OnSets),Immutable);
#Add(bl,Immutable(Set(b)));
op:=ActionHomomorphism(G,Set(b),OnSets,"surjective");
ImagesSource(op:onlyimage); #`onlyimage' forces same generators
k:=KernelOfMultiplicativeGeneralMapping(op);
if onlykernel<>fail and k=onlykernel and Length(b)<found then
found:=Length(b);
fi;
AddNaturalHomomorphismsPool(G,k,op);
# also one finer blocks (to make up for iterating only once)
if type=2 then
newb:=Blocks(G,b,OnSets);
else
newb:=Blocks(Go,Blocks(Go,mov,j),OnSets);
fi;
if Length(newb)>1 then
newb:=Union(newb[1]);
if not (newb in allb or newb in allbold) then
Add(allb,newb);
SortBy(allb,Length);
fi;
fi;
fi;
od;
#if Length(i)<500 and Size(Go)>10*Length(i) then
#else
# # one block system
# b:=Blocks(G,i);
# if Length(b)>1 then
# Add(bl,Immutable(Set(b)));
# fi;
# fi;
od;
pool.GopDone:=true;
fi;
end);
BindGlobal("DoActionBlocksForKernel",
function(G,mustfaithful)
local dom, o, bl, j, b, allb,newb;
dom:=MovedPoints(G);
# orbits
o:=OrbitsDomain(G,dom);
o:=Set(o,Set);
# all good blocks
bl:=dom;
allb:=ShallowCopy(RepresentativesMinimalBlocks(G,dom));
for j in allb do
if Length(dom)/Length(j)<Length(bl) and
Size(Core(mustfaithful,Stabilizer(mustfaithful,j,OnSets)))=1
then
b:=Orbit(G,j,OnSets);
bl:=b;
# also one finer blocks (as we iterate only once)
newb:=Blocks(G,b,OnSets);
if Length(newb)>1 then
newb:=Union(newb[1]);
if not newb in allb then
Add(allb,newb);
fi;
fi;
fi;
od;
if Length(bl)<Length(dom) then
return bl;
else
return fail;
fi;
end);
#############################################################################
##
#F GenericFindActionKernel random search for subgroup with faithful core
##
BADINDEX:=1000; # the index that is too big
BindGlobal( "GenericFindActionKernel", function(arg)
local G, N, knowi, goodi, simple, uc, zen, cnt, pool, ise, v, badi,
totalcnt, interrupt, u, nu, cor, zzz,bigperm,perm,badcores,max,i,hard;
G:=arg[1];
N:=arg[2];
if Length(arg)>2 then
knowi:=arg[3];
else
knowi:=IndexNC(G,N);
fi;
# special treatment for solvable groups. This will never be triggered for
# perm groups or nice groups
if Size(N)>1 and HasSolvableFactorGroup(G,N) then
perm:=ActionHomomorphism(G,RightCosets(G,N),OnRight,"surjective");
perm:=perm*IsomorphismPcGroup(Image(perm));
return perm;
fi;
# special treatment for abelian factor
if HasAbelianFactorGroup(G,N) then
if IsPermGroup(G) and Size(N)=1 then
return IdentityMapping(G);
else
perm:=ActionHomomorphism(G,RightCosets(G,N),OnRight,"surjective");
fi;
return perm;
fi;
bigperm:=IsPermGroup(G) and NrMovedPoints(G)>10000;
# what is a good degree:
goodi:=Minimum(Int(knowi*9/10),LogInt(IndexNC(G,N),2)^2);
simple:=HasIsNonabelianSimpleGroup(G) and IsNonabelianSimpleGroup(G) and Size(N)=2;
uc:=TrivialSubgroup(G);
# look if it is worth to look at action on N
# if not abelian: later replace by abelian Normal subgroup
if IsAbelian(N) and (Size(N)>50 or IndexNC(G,N)<Factorial(Size(N)))
and Size(N)<50000 then
zen:=Centralizer(G,N);
if Size(zen)=Size(N) then
cnt:=0;
repeat
cnt:=cnt+1;
zen:=Centralizer(G,Random(N));
if (simple or Size(Core(G,zen))=Size(N)) and
IndexNC(G,zen)<IndexNC(G,uc) then
uc:=zen;
fi;
# until enough searched or just one orbit
until cnt=9 or (IndexNC(G,zen)+1=Size(N));
AddNaturalHomomorphismsPool(G,N,uc,IndexNC(G,uc));
else
Info(InfoFactor,3,"centralizer too big");
fi;
fi;
pool:=NaturalHomomorphismsPool(G);
pool.dotriv:=true;
CloseNaturalHomomorphismsPool(G,N);
pool.dotriv:=false;
ise:=Filtered(pool.ker,x->IsSubset(x,N));
if Length(ise)=0 then
ise:=G;
else
ise:=Intersection(ise);
fi;
# try a random extension step
# (We might always first add a random element and get something bigger)
v:=N;
#if Length(arg)=3 then
## in one example 512->90, ca. 40 tries
#cnt:=Int(arg[3]/10);
#else
#cnt:=25;
#fi;
badcores:=[];
badi:=BADINDEX;
hard:=ValueOption("hard");
if hard=fail then
hard:=100000;
elif hard=true then
hard:=10000;
fi;
totalcnt:=0;
interrupt:=false;
cnt:=20;
repeat
u:=v;
repeat
repeat
if Length(arg)<4 or Random(1,2)=1 then
if IsCyclic(u) and Random(1,4)=1 then
# avoid being stuck with a bad first element
u:=Subgroup(G,[Random(G)]);
fi;
if Length(GeneratorsOfGroup(u))<2 then
# closing might cost a big stabilizer chain calculation -- just
# recreate
nu:=Group(Concatenation(GeneratorsOfGroup(u),[Random(G)]));
else
nu:=ClosureGroup(u,Random(G));
fi;
else
if Length(GeneratorsOfGroup(u))<2 then
# closing might cost a big stabilizer chain calculation -- just
# recreate
nu:=Group(Concatenation(GeneratorsOfGroup(u),[Random(arg[4])]));
else
nu:=ClosureGroup(u,Random(arg[4]));
fi;
fi;
SetParent(nu,G);
totalcnt:=totalcnt+1;
if KnownNaturalHomomorphismsPool(G,N) and
Minimum(IndexNC(G,v),knowi)<hard
and 5*totalcnt>Minimum(IndexNC(G,v),knowi,1000) then
# interrupt if we're already quite good
interrupt:=true;
fi;
if ForAny(badcores,x->IsSubset(nu,x)) then
nu:=u;
fi;
# Abbruchkriterium: Bis kein Normalteiler, es sei denn, es ist N selber
# (das brauchen wir, um in einigen trivialen F"allen abbrechen zu
# k"onnen)
#Print("nu=",Length(GeneratorsOfGroup(nu))," : ",Size(nu),"\n");
until
# der Index ist nicht so klein, da"s wir keine Chance haben
((not bigperm or
Length(Orbit(nu,MovedPoints(G)[1]))<NrMovedPoints(G)) and
(IndexNC(G,nu)>50 or Factorial(IndexNC(G,nu))>=IndexNC(G,N)) and
not IsNormal(G,nu)) or IsSubset(u,nu) or interrupt;
Info(InfoFactor,4,"Index ",IndexNC(G,nu));
u:=nu;
until totalcnt>300 or
# und die Gruppe ist nicht zuviel schlechter als der
# beste bekannte Index. Daf"ur brauchen wir aber wom"oglich mehrfache
# Erweiterungen.
interrupt or (((Length(arg)=2 or IndexNC(G,u)<knowi)));
if IndexNC(G,u)<knowi then
#Print("Index:",IndexNC(G,u),"\n");
if simple and u<>G then
cor:=TrivialSubgroup(G);
else
cor:=Core(G,u);
fi;
if Size(cor)>Size(N) and IsSubset(cor,N) and not cor in badcores then
Add(badcores,cor);
fi;
# store known information(we do't act, just store the subgroup).
# Thus this is fairly cheap
pool.dotriv:=true;
zzz:=AddNaturalHomomorphismsPool(G,cor,u,IndexNC(G,u));
if IsPerm(zzz) and zzz<>() then
CloseNaturalHomomorphismsPool(G,N);
fi;
pool.dotriv:=false;
zzz:=DegreeNaturalHomomorphismsPool(G,N);
Info(InfoFactor,3," ext ",cnt,": ",IndexNC(G,u)," best degree:",zzz);
if cnt<10 and Size(cor)>Size(N) and IndexNC(G,u)*2<knowi and
ValueOption("inmax")=fail then
if IsSubset(SolvableRadical(u),N) and Size(N)<Size(SolvableRadical(u)) then
# only affine ones are needed, rest will have wrong kernel
max:=DoMaxesTF(u,["1"]:inmax,cheap);
else
max:=TryMaximalSubgroupClassReps(u:inmax,cheap);
fi;
max:=Filtered(max,x->IndexNC(G,x)<knowi and IsSubset(x,N));
for i in max do
cor:=Core(G,i);
AddNaturalHomomorphismsPool(G,cor,i,IndexNC(G,i));
od;
zzz:=DegreeNaturalHomomorphismsPool(G,N);
Info(InfoFactor,3," Maxes: ",Length(max)," best degree:",zzz);
fi;
else
zzz:=DegreeNaturalHomomorphismsPool(G,N);
fi;
if IsInt(zzz) then
knowi:=zzz;
fi;
cnt:=cnt-1;
if cnt=0 and zzz>badi then
badi:=Int(badi*12/10);
Info(InfoWarning+InfoFactor,2,
"index unreasonably large, iterating ",badi);
cnt:=20;
totalcnt:=0;
interrupt:=false;
v:=N; # all new
fi;
until interrupt or cnt<=0 or zzz<=goodi;
Info(InfoFactor,1,zzz," vs ",badi);
return GetNaturalHomomorphismsPool(G,N);
end );
#############################################################################
##
#F SmallerDegreePermutationRepresentation( <G> )
##
InstallGlobalFunction(SmallerDegreePermutationRepresentation,function(G)
local o, s, k, gut, erg, H, hom, b, ihom, improve, map, loop,bl,
i,cheap,k2,change;
change:=false;
Info(InfoFactor,1,"Smaller degree for order ",Size(G),", deg: ",NrMovedPoints(G));
cheap:=ValueOption("cheap");
if cheap="skip" then
return IdentityMapping(G);
fi;
cheap:=cheap=true;
if Length(GeneratorsOfGroup(G))>7 then
s:=SmallGeneratingSet(G);
if Length(s)=0 then s:=[One(G)];fi;
if Length(s)<Length(GeneratorsOfGroup(G))-1 then
Info(InfoFactor,1,"reduced to ",Length(s)," generators");
H:=Group(s);
change:=true;
SetSize(H,Size(G));
return SmallerDegreePermutationRepresentation(H);
fi;
fi;
# deal with large abelian components first (which could be direct)
if cheap<>true then
hom:=MaximalAbelianQuotient(G);
i:=IndependentGeneratorsOfAbelianGroup(Image(hom));
o:=List(i,Order);
if ValueOption("norecurse")<>true and
Product(o)>20 and Sum(o)*4<NrMovedPoints(G) then
Info(InfoFactor,2,"append abelian rep");
s:=AbelianGroup(IsPermGroup,o);
ihom:=GroupHomomorphismByImagesNC(Image(hom),s,i,GeneratorsOfGroup(s));
erg:=SubdirectDiagonalPerms(
List(GeneratorsOfGroup(G),x->Image(ihom,Image(hom,x))),
GeneratorsOfGroup(G));
k:=Group(erg);SetSize(k,Size(G));
hom:=GroupHomomorphismByImagesNC(G,k,GeneratorsOfGroup(G),erg);
return hom*SmallerDegreePermutationRepresentation(k:norecurse);
fi;
fi;
# known simple?
if HasIsSimpleGroup(G) and IsSimpleGroup(G)
and NrMovedPoints(G)>=SufficientlySmallDegreeSimpleGroupOrder(Size(G))
then return IdentityMapping(G);
fi;
if not IsTransitive(G,MovedPoints(G)) then
o:=ShallowCopy(OrbitsDomain(G,MovedPoints(G)));
SortBy(o, Length);
for loop in [1..2] do
s:=[];
# Try subdirect product
k:=G;
gut:=[];
for i in [1..Length(o)] do
s:=Stabilizer(k,o[i],OnTuples);
if Size(s)<Size(k) then
k:=s;
Add(gut,i);
fi;
od;
# reduce each orbit separately
o:=o{gut};
# second run: now take the big orbits first
Sort(o,function(a,b)return Length(a)>Length(b);end);
od;
SortBy(o, Length);
erg:=List(GeneratorsOfGroup(G),i->());
k:=G;
for i in [1..Length(o)] do
Info(InfoFactor,1,"Try to shorten orbit ",i," Length ",Length(o[i]));
s:=ActionHomomorphism(G,o[i],OnPoints,"surjective");
k2:=Image(s,k);
k:=Stabilizer(k,o[i],OnTuples);
H:=Range(s);
# is there an action that is good enough for improving the overall
# kernel, even if it is not faithful? If so use the best of them.
b:=DoActionBlocksForKernel(H,k2);
if b<>fail then
Info(InfoFactor,2,"Blocks for kernel reduce to ",Length(b));
b:=ActionHomomorphism(H,b,OnSets,"surjective");
s:=s*b;
fi;
s:=s*SmallerDegreePermutationRepresentation(Image(s));
Info(InfoFactor,1,"Shortened to ",NrMovedPoints(Range(s)));
erg:=SubdirectDiagonalPerms(erg,List(GeneratorsOfGroup(G),i->Image(s,i)));
od;
if NrMovedPoints(erg)<NrMovedPoints(G) then
s:=Group(erg,()); # `erg' arose from `SubdirectDiagonalPerms'
SetSize(s,Size(G));
s:=GroupHomomorphismByImagesNC(G,s,GeneratorsOfGroup(G),erg);
SetIsBijective(s,true);
return s;
fi;
return IdentityMapping(G);
fi; # intransitive treatment
# if the original group has no stabchain we probably do not want to keep
# it (or a homomorphisms pool) there -- make a copy for working
# intermediately with it.
if not HasStabChainMutable(G) then
H:= GroupWithGenerators( GeneratorsOfGroup( G ),One(G) );
change:=true;
if HasSize(G) then
SetSize(H,Size(G));
fi;
if HasBaseOfGroup(G) then
SetBaseOfGroup(H,BaseOfGroup(G));
fi;
else
H:=G;
fi;
hom:=IdentityMapping(H);
b:=NaturalHomomorphismsPool(H);
b.dotriv:=true;
AddNaturalHomomorphismsPool(H,TrivialSubgroup(H),hom,NrMovedPoints(H));
b.dotriv:=false;
# cheap initial block reduction?
if IsTransitive(H,MovedPoints(H)) then
improve:=true;
while improve and (cheap or NrMovedPoints(H)*5>Size(H)) do
improve:=false;
bl:=Blocks(H,MovedPoints(H));
map:=ActionHomomorphism(G,bl,OnSets,"surjective");
ImagesSource(map:onlyimage); #`onlyimage' forces same generators
bl:=KernelOfMultiplicativeGeneralMapping(map);
AddNaturalHomomorphismsPool(G,bl,map);
if Size(bl)=1 then
hom:=hom*map;
H:=Image(map);
change:=true;
Info(InfoFactor,2," quickblocks improved to degree ",NrMovedPoints(H));
fi;
od;
fi;
b:=NaturalHomomorphismsPool(H);
b.dotriv:=true;
if change then
DoCheapActionImages(H:onlykernel:=TrivialSubgroup(H));
else
DoCheapActionImages(H);
fi;
CloseNaturalHomomorphismsPool(H,TrivialSubgroup(H));
b.dotriv:=false;
map:=GetNaturalHomomorphismsPool(H,TrivialSubgroup(H));
if map<>fail and Image(map)<>H then
Info(InfoFactor,2,"cheap actions improved to degree ",NrMovedPoints(H));
hom:=hom*map;
H:=Image(map);
fi;
o:=DegreeNaturalHomomorphismsPool(H,TrivialSubgroup(H));
if cheap<>true and (IsBool(o) or o*2>=NrMovedPoints(H)) then
s:=GenericFindActionKernel(H,TrivialSubgroup(H),NrMovedPoints(H));
if s<>fail then
hom:=hom*s;
fi;
fi;
return hom;
end);
#############################################################################
##
#F ImproveActionDegreeByBlocks( <G>, <N> , hom )
## extension of <U> in <G> such that \bigcap U^g=N remains valid
##
InstallGlobalFunction(ImproveActionDegreeByBlocks,function(G,N,oh)
local gimg,img,dom,b,improve,bp,bb,i,k,bestdeg,subo,op,bc,bestblock,bdom,
bestop,sto,subomax;
Info(InfoFactor,1,"try to find block systems");
# remember that we computed the blocks
b:=NaturalHomomorphismsPool(G);
# special case to use it for improving a permutation representation
if Size(N)=1 then
Info(InfoFactor,1,"special case for trivial subgroup");
b.ker:=[N];
b.ops:=[oh];
b.cost:=[Length(MovedPoints(Range(oh)))];
b.lock:=[false];
b.blocksdone:=[false];
subomax:=20;
else
subomax:=500;
fi;
i:=PositionSet(b.ker,N);
if b.blocksdone[i] then
return DegreeNaturalHomomorphismsPool(G,N); # we have done it already
fi;
b.blocksdone[i]:=true;
if not IsPermGroup(Range(oh)) then
return 1;
fi;
gimg:=Image(oh,G);
img:=gimg;
dom:=MovedPoints(img);
bdom:=fail;
if IsTransitive(img,dom) then
# one orbit: Blocks
repeat
b:=Blocks(img,dom);
improve:=false;
if Length(b)>1 then
if Length(dom)<40000 then
subo:=ApproximateSuborbitsStabilizerPermGroup(img,dom[1]);
subo:=Difference(List(subo,i->i[1]),dom{[1]});
else
subo:=fail;
fi;
bc:=First(b,i->dom[1] in i);
if subo<>fail and (Length(subo)<=subomax) then
Info(InfoFactor,2,"try all seeds");
# if the degree is not too big or if we are desperate then go for
# all blocks
# greedy approach: take always locally best one (otherwise there
# might be too much work to do)
bestdeg:=Length(dom);
bp:=[]; #Blocks pool
i:=1;
while i<=Length(subo) do
if subo[i] in bc then
bb:=b;
else
bb:=Blocks(img,dom,[dom[1],subo[i]]);
fi;
if Length(bb)>1 and not (bb[1] in bp or Length(bb)>bestdeg) then
Info(InfoFactor,3,"found block system ",Length(bb));
# new nontriv. system found
AddSet(bp,bb[1]);
# store action
op:=1;# remove old homomorphism to free memory
if bdom<>fail then
bb:=Set(bb,i->Immutable(Union(bdom{i})));
fi;
op:=ActionHomomorphism(gimg,bb,OnSets,"surjective");
if HasSize(gimg) and not HasStabChainMutable(gimg) then
sto:=StabChainOptions(Range(op));
sto.limit:=Size(gimg);
# try only with random (will exclude some chances, but is
# quicker. If the size is OK we have a proof anyhow).
sto.random:=100;
# if gimgbas<>false then
# SetBaseOfGroup(Range(op),
# List(gimgbas,i->PositionProperty(bb,j->i in j)));
# fi;
if Size(Range(op))=Size(gimg) then
sto.random:=1000;
k:=TrivialSubgroup(gimg);
op:=oh*op;
SetKernelOfMultiplicativeGeneralMapping(op,PreImage(oh,k));
AddNaturalHomomorphismsPool(G,
KernelOfMultiplicativeGeneralMapping(op),
op,Length(bb));
else
k:=[]; # do not trigger improvement
fi;
else
k:=KernelOfMultiplicativeGeneralMapping(op);
SetSize(Range(op),IndexNC(gimg,k));
op:=oh*op;
SetKernelOfMultiplicativeGeneralMapping(op,PreImage(oh,k));
AddNaturalHomomorphismsPool(G,
KernelOfMultiplicativeGeneralMapping(op),
op,Length(bb));
fi;
# and note whether we got better
#improve:=improve or (Size(k)=1);
if Size(k)=1 and Length(bb)<bestdeg then
improve:=true;
bestdeg:=Length(bb);
bestblock:=bb;
bestop:=op;
fi;
fi;
# break the test loop if we found a fairly small block system
# (iterate greedily immediately)
if improve and bestdeg<i then
i:=Length(dom);
fi;
i:=i+1;
od;
else
Info(InfoFactor,2,"try only one system");
op:=1;# remove old homomorphism to free memory
if bdom<>fail then
b:=Set(b,i->Immutable(Union(bdom{i})));
fi;
op:=ActionHomomorphism(gimg,b,OnSets,"surjective");
if HasSize(gimg) and not HasStabChainMutable(gimg) then
sto:=StabChainOptions(Range(op));
sto.limit:=Size(gimg);
# try only with random (will exclude some chances, but is
# quicker. If the size is OK we have a proof anyhow).
sto.random:=100;
# if gimgbas<>false then
# SetBaseOfGroup(Range(op),
# List(gimgbas,i->PositionProperty(b,j->i in j)));
# fi;
if Size(Range(op))=Size(gimg) then
sto.random:=1000;
k:=TrivialSubgroup(gimg);
op:=oh*op;
SetKernelOfMultiplicativeGeneralMapping(op,PreImage(oh,k));
AddNaturalHomomorphismsPool(G,
KernelOfMultiplicativeGeneralMapping(op),
op,Length(b));
else
k:=[]; # do not trigger improvement
fi;
else
k:=KernelOfMultiplicativeGeneralMapping(op);
SetSize(Range(op),IndexNC(gimg,k));
# keep action knowledge
op:=oh*op;
SetKernelOfMultiplicativeGeneralMapping(op,PreImage(oh,k));
AddNaturalHomomorphismsPool(G,
KernelOfMultiplicativeGeneralMapping(op),
op,Length(b));
fi;
if Size(k)=1 then
improve:=true;
bestblock:=b;
bestop:=op;
fi;
fi;
if improve then
# update mapping
bdom:=bestblock;
img:=Image(bestop,G);
dom:=MovedPoints(img);
fi;
fi;
until improve=false;
fi;
Info(InfoFactor,1,"end of blocks search");
return DegreeNaturalHomomorphismsPool(G,N);
end);
#############################################################################
##
#M FindActionKernel(<G>) . . . . . . . . . . . . . . . . . . . . generic
##
InstallMethod(FindActionKernel,"generic for finite groups",IsIdenticalObj,
[IsGroup and IsFinite,IsGroup],0,GenericFindActionKernel);
RedispatchOnCondition(FindActionKernel,IsIdenticalObj,[IsGroup,IsGroup],
[IsGroup and IsFinite,IsGroup],0);
InstallMethod(FindActionKernel,"general case: can't do",IsIdenticalObj,
[IsGroup,IsGroup],0,ReturnFail);
BindGlobal("FactPermRepMaxDesc",function(g,n,maxlev)
local lim,deg,all,c,recurse,use,start;
if ValueOption("infactorpermrep")=true then return false;fi;
deg:=DegreeNaturalHomomorphismsPool(g,n);
if deg=fail then deg:=infinity;fi;
all:=[];
start:=ClosureGroup(DerivedSubgroup(g),n);
lim:=RootInt(IndexNC(g,n),3)*Maximum(1,LogInt(IndexNC(g,start),2));
c:=start;
Info(InfoFactor,1,"Try maximals for limit ",lim," from ",deg);
recurse:=function(a,lev)
local m,ma,nm,i,j,co,wait,use;
Info(InfoFactor,3,"pop in ",lev);
m:=[a];
while Length(m)>0 do
wait:=[];
ma:=[];
for i in m do
if ForAll(all,y->RepresentativeAction(g,i,y)=fail) then
Add(all,i);
Info(InfoFactor,2,"Maximals of index ",IndexNC(g,i));
nm:=TryMaximalSubgroupClassReps(i:inmax,infactorpermrep,cheap);
nm:=Filtered(nm,x->IndexNC(g,x)<=lim and IsSubset(x,n) and not
IsNormal(g,x));
for j in nm do
if IsSubset(j,c) then
use:=ClosureGroup(n,DerivedSubgroup(j));
if not IsSubset(use,c) then
j:=use;
use:=true;
else
Add(wait,j);
use:=false;
fi;
else
use:=true;
fi;
if use then
co:=Core(g,j);
AddNaturalHomomorphismsPool(g,co,j,IndexNC(g,j));
c:=Intersection(co,c);
Add(ma,j);
fi;
od;
else
Info(InfoFactor,2,"discard conjugate");
fi;
od;
if Length(ma)>0 then
CloseNaturalHomomorphismsPool(g,n);
i:=DegreeNaturalHomomorphismsPool(g,n);
if i<deg then
deg:=i;
Info(InfoFactor,1,"Itmax improves to degree ",deg);
if lev>1 or deg<lim then return true;fi;
fi;
m:=ma;
SortBy(m,x->-Size(x));
elif lev<maxlev then
# no improvement. Go down
wait:=Filtered(wait,x->IndexNC(g,x)*10<=lim);
for i in wait do
if recurse(i,lev+1) then return true;fi;
od;
m:=[];
else
m:=[];
fi;
od;
if Size(c)>Size(n) then
Info(InfoFactor,3,"pop up failure ",Size(c));
else
Info(InfoFactor,3,"pop up found ",Size(c));
fi;
return false;
end;
return recurse(start,1);
end);
#############################################################################
##
#M FindActionKernel(<G>) . . . . . . . . . . . . . . . . . . . . permgrp
##
InstallMethod(FindActionKernel,"perm",IsIdenticalObj,
[IsPermGroup,IsPermGroup],0,
function(G,N)
local pool, dom, bestdeg, blocksdone, o, s, badnormals, cnt, v, u, oo, m,
badcomb, idx, i, comb,act,k,j;
if IndexNC(G,N)<50 then
# small index, anything is OK
return GenericFindActionKernel(G,N);
else
# get the known ones, including blocks &c. which might be of use
DoCheapActionImages(G);
# find smallish layer actions
oo:=ClosureGroup(SolvableRadical(G),N);
dom:=ChiefSeriesThrough(G,[oo,N]);
dom:=Filtered(dom,x->IsSubset(oo,x) and IsSubset(x,N));
i:=2;
while i<=Length(dom) do
j:=i;
while j<Length(dom)
#and HasElementaryAbelianFactorGroup(dom[i-1],dom[j+1])
and IndexNC(dom[i-1],dom[j+1])<=2000 do
j:=j+1;
od;
if IndexNC(dom[i-1],dom[j])<=2000 then
v:=RightTransversal(dom[i-1],dom[j]);
oo:=OrbitsDomain(G,v,function(rep,g)
return v[PositionCanonical(v,rep^g)];
end);
for k in oo do
if Length(k)>1 then
u:=Stabilizer(G,k[1],function(x,g)
return v[PositionCanonical(v,x^g)];
end);
repeat
if not IsNormal(G,u) then
AddNaturalHomomorphismsPool(G,Core(G,u),u,IndexNC(G,u));
fi;
m:=u;
u:=ClosureGroup(N,DerivedSubgroup(u));
until m=u;
fi;
od;
fi;
i:=j+1;
od;
pool:=NaturalHomomorphismsPool(G);
dom:=MovedPoints(G);
# store regular to have one anyway
bestdeg:=IndexNC(G,N);
AddNaturalHomomorphismsPool(G,N,N,bestdeg);
# check if there are multiple orbits
o:=Orbits(G,MovedPoints(G));
s:=List(o,i->Stabilizer(G,i,OnTuples));
if not ForAny(s,i->IsSubset(N,i)) then
Info(InfoFactor,2,"Try reduction to orbits");
s:=List(s,i->ClosureGroup(i,N));
if Intersection(s)=N then
Info(InfoFactor,1,"Reduction to orbits will do");
List(s,i->NaturalHomomorphismByNormalSubgroup(G,i));
fi;
fi;
CloseNaturalHomomorphismsPool(G,N);
# action in orbit image -- sometimes helps
if Length(o)>1 then
for i in o do
act:=ActionHomomorphism(G,i,OnPoints,"surjective");
k:=KernelOfMultiplicativeGeneralMapping(act);
k:=ClosureGroup(k,N); # pre-image of (image of normal subgroup under act)
u:=Image(act,N);
v:=NaturalHomomorphismByNormalSubgroupNC(Image(act),u);
o:=DegreeNaturalHomomorphismsPool(Image(act),u);
if IsInt(o) then # otherwise its solvable factor we do differently
AddNaturalHomomorphismsPool(G,k,act*v,o);
fi;
od;
CloseNaturalHomomorphismsPool(G,N);
fi;
bestdeg:=DegreeNaturalHomomorphismsPool(G,N);
Info(InfoFactor,1,"Orbits and known, best Index ",bestdeg);
blocksdone:=false;
# use subgroup that fixes a base of N
# get orbits of a suitable stabilizer.
o:=BaseOfGroup(N);
s:=Stabilizer(G,o,OnTuples);
badnormals:=Filtered(pool.ker,i->IsSubset(i,N) and Size(i)>Size(N));
if Size(s)>1 and IndexNC(G,s)/Size(N)<2000 and bestdeg>IndexNC(G,s) then
cnt:=Filtered(OrbitsDomain(s,dom),i->Length(i)>1);
for i in cnt do
v:=ClosureGroup(N,Stabilizer(s,i[1]));
if Size(v)>Size(N) and IndexNC(G,v)<2000
and not ForAny(badnormals,j->IsSubset(v,j)) then
u:=Core(G,v);
if Size(u)>Size(N) and IsSubset(u,N) and not u in badnormals then
Add(badnormals,u);
fi;
AddNaturalHomomorphismsPool(G,u,v,IndexNC(G,v));
fi;
od;
# try also intersections
CloseNaturalHomomorphismsPool(G,N);
bestdeg:=DegreeNaturalHomomorphismsPool(G,N);
Info(InfoFactor,1,"Base Stabilizer and known, best Index ",bestdeg);
if bestdeg<500 and bestdeg<IndexNC(G,N) then
# should be better...
bestdeg:=ImproveActionDegreeByBlocks(G,N,
GetNaturalHomomorphismsPool(G,N));
blocksdone:=true;
Info(InfoFactor,2,"Blocks improve to ",bestdeg);
fi;
fi;
# then we should look at the orbits of the normal subgroup to see,
# whether anything stabilizing can be of use
o:=Filtered(OrbitsDomain(N,dom),i->Length(Orbit(G,i[1]))>Length(i));
Apply(o,Set);
oo:=OrbitsDomain(G,o,OnSets);
s:=G;
for i in oo do
s:=StabilizerOfBlockNC(s,i[1]);
od;
Info(InfoFactor,2,"stabilizer of index ",IndexNC(G,s));
if not ForAny(badnormals,j->IsSubset(s,j)) then
m:=Core(G,s); # the normal subgroup we get this way.
if Size(m)>Size(N) and IsSubset(m,N) and not m in badnormals then
Add(badnormals,m);
fi;
AddNaturalHomomorphismsPool(G,m,s,IndexNC(G,s));
else
m:=G; # guaranteed fail
fi;
if Size(m)=Size(N) and IndexNC(G,s)<bestdeg then
bestdeg:=IndexNC(G,s);
blocksdone:=false;
Info(InfoFactor,2,"Orbits Stabilizer improves to index ",bestdeg);
elif Size(m)>Size(N) then
# no hard work for trivial cases
if 2*IndexNC(G,N)>Length(o) then
# try to find a subgroup, which does not contain any part of m
# For wreath products (the initial aim), the following method works
# fairly well
v:=Subgroup(G,Filtered(GeneratorsOfGroup(G),i->not i in m));
v:=SmallGeneratingSet(v);
cnt:=1;
badcomb:=[];
repeat
Info(InfoFactor,3,"Trying",cnt);
for comb in Combinations([1..Length(v)],cnt) do
#Print(">",comb,"\n");
if not ForAny(badcomb,j->IsSubset(comb,j)) then
u:=SubgroupNC(G,v{comb});
o:=ClosureGroup(N,u);
idx:=Size(G)/Size(o);
if idx<10 and Factorial(idx)*Size(N)<Size(G) then
# the permimage won't be sufficiently large
AddSet(badcomb,Immutable(comb));
fi;
if idx<bestdeg and Size(G)>Size(o)
and not ForAny(badnormals,i->IsSubset(o,i)) then
m:=Core(G,o);
if Size(m)>Size(N) and IsSubset(m,N) then
Info(InfoFactor,3,"Core ",comb," failed");
AddSet(badcomb,Immutable(comb));
if not m in badnormals then
Add(badnormals,m);
fi;
fi;
if idx<bestdeg and Size(m)=Size(N) then
Info(InfoFactor,3,"Core ",comb," succeeded");
bestdeg:=idx;
AddNaturalHomomorphismsPool(G,N,o,bestdeg);
blocksdone:=false;
cnt:=0;
fi;
fi;
fi;
od;
cnt:=cnt+1;
until cnt>Length(v);
fi;
fi;
Info(InfoFactor,2,"Orbits Stabilizer, Best Index ",bestdeg);
# first force blocks
if (not blocksdone) and bestdeg<200 and bestdeg<IndexNC(G,N) then
Info(InfoFactor,3,"force blocks");
bestdeg:=ImproveActionDegreeByBlocks(G,N,
GetNaturalHomomorphismsPool(G,N));
blocksdone:=true;
Info(InfoFactor,2,"Blocks improve to ",bestdeg);
fi;
if bestdeg=IndexNC(G,N) or
(bestdeg>400 and not(bestdeg<=2*NrMovedPoints(G))) then
if GenericFindActionKernel(G,N,bestdeg,s)<>fail then
blocksdone:=true;
fi;
bestdeg:=DegreeNaturalHomomorphismsPool(G,N);
Info(InfoFactor,1," Random search found ",bestdeg);
fi;
if bestdeg>10000 and bestdeg^2>IndexNC(G,N) then
cnt:=bestdeg;
FactPermRepMaxDesc(G,N,5);
bestdeg:=DegreeNaturalHomomorphismsPool(G,N);
if bestdeg<cnt then blocksdone:=false;fi;
Info(InfoFactor,1,"Iterated maximals found ",bestdeg);
fi;
if not blocksdone then
ImproveActionDegreeByBlocks(G,N,GetNaturalHomomorphismsPool(G,N));
fi;
Info(InfoFactor,3,"return hom");
return GetNaturalHomomorphismsPool(G,N);
return o;
fi;
end);
#############################################################################
##
#M FindActionKernel(<G>) . . . . . . . . . . . . . . . . . . . . generic
##
InstallMethod(FindActionKernel,"Niceo",IsIdenticalObj,
[IsGroup and IsHandledByNiceMonomorphism,IsGroup],0,
function(G,N)
local hom,hom2;
hom:=NiceMonomorphism(G);
hom2:=GenericFindActionKernel(NiceObject(G),Image(hom,N));
if hom2<>fail then
return hom*hom2;
else
return hom;
fi;
end);
BindGlobal("FACTGRP_TRIV",Group([],()));
#############################################################################
##
#M NaturalHomomorphismByNormalSubgroup( <G>, <N> ) . . mapping G ->> G/N
## this function returns an epimorphism from G
## with kernel N. The range of this mapping is a suitable (isomorphic)
## permutation group (with which we can compute much easier).
InstallMethod(NaturalHomomorphismByNormalSubgroupOp,
"search for operation",IsIdenticalObj,[IsGroup,IsGroup],0,
function(G,N)
local proj,h,pool;
# catch the trivial case N=G
if CanComputeIndex(G,N) and IndexNC(G,N)=1 then
h:=FACTGRP_TRIV; # a new group is created
h:=GroupHomomorphismByImagesNC( G, h, GeneratorsOfGroup( G ),
List( GeneratorsOfGroup( G ), i -> () )); # a new group is created
SetKernelOfMultiplicativeGeneralMapping( h, G );
return h;
fi;
# catch trivial case N=1 (IsTrivial might not be set)
if (HasSize(N) and Size(N)=1) or (HasGeneratorsOfGroup(N) and
ForAll(GeneratorsOfGroup(N),IsOne)) then
return IdentityMapping(G);
fi;
# check, whether we already know a factormap
pool:=NaturalHomomorphismsPool(G);
h:=PositionSet(pool.ker,N);
if h<>fail and IsGeneralMapping(pool.ops[h]) then
return GetNaturalHomomorphismsPool(G,N);
fi;
DoCheapActionImages(G);
if HasSolvableRadical(G) and N=SolvableRadical(G) then
h:=GetNaturalHomomorphismsPool(G,N);
fi;
if HasDirectProductInfo(G) and DegreeNaturalHomomorphismsPool(G,N)=fail then
for proj in [1..Length(DirectProductInfo(G).groups)] do
proj:=Projection(G,proj);
h:=NaturalHomomorphismByNormalSubgroup(Image(proj,G),Image(proj,N));
AddNaturalHomomorphismsPool(G,
ClosureGroup(KernelOfMultiplicativeGeneralMapping(proj),N),proj*h);
od;
fi;
CloseNaturalHomomorphismsPool(G,N);
h:=DegreeNaturalHomomorphismsPool(G,N);
if h<>fail and RootInt(h^3,2)<IndexNC(G,N) then
h:=GetNaturalHomomorphismsPool(G,N);
else
h:=fail;
fi;
if h=fail then
# now we try to find a suitable operation
# redispatch upon finiteness test, as following will fail in infinite case
if not HasIsFinite(G) and IsFinite(G) then
return NaturalHomomorphismByNormalSubgroupOp(G,N);
fi;
h:=FindActionKernel(G,N);
if h<>fail then
Info(InfoFactor,1,"Action of degree ",
Length(MovedPoints(Range(h)))," found");
else
Error("I don't know how to find a natural homomorphism for <N> in <G>");
# nothing had been found, Desperately one could try again, but that
# would create a possible infinite loop.
h:= NaturalHomomorphismByNormalSubgroup( G, N );
fi;
fi;
# return the map
return h;
end);
RedispatchOnCondition(NaturalHomomorphismByNormalSubgroupNCOrig,IsIdenticalObj,
[IsGroup,IsGroup],[IsGroup and IsFinite,IsGroup],0);
RedispatchOnCondition(NaturalHomomorphismByNormalSubgroupInParent,true,
[IsGroup],[IsGroup and IsFinite],0);
RedispatchOnCondition(FactorGroupNC,IsIdenticalObj,
[IsGroup,IsGroup],[IsGroup and IsFinite,IsGroup],0);
#############################################################################
##
#M NaturalHomomorphismByNormalSubgroup( <G>, <N> ) . . for solvable factors
##
NH_TRYPCGS_LIMIT:=30000;
InstallMethod( NaturalHomomorphismByNormalSubgroupOp,
"test if known/try solvable factor for permutation groups",
IsIdenticalObj, [ IsPermGroup, IsPermGroup ], 0,
function( G, N )
local map, pcgs, A, filter;
if KnownNaturalHomomorphismsPool(G,N) then
A:=DegreeNaturalHomomorphismsPool(G,N);
if A<50 or (IsInt(A) and A<IndexNC(G,N)/LogInt(IndexNC(G,N),2)^2) then
map:=GetNaturalHomomorphismsPool(G,N);
if map<>fail then
Info(InfoFactor,2,"use stored map");
return map;
fi;
fi;
fi;
if IndexNC(G,N)=1 or Size(N)=1
or Minimum(IndexNC(G,N),NrMovedPoints(G))>NH_TRYPCGS_LIMIT then
TryNextMethod();
fi;
# Make a pcgs based on an elementary abelian series (good for ag
# routines).
pcgs := TryPcgsPermGroup( [ G, N ], false, false, true );
if not IsModuloPcgs( pcgs ) then
TryNextMethod();
fi;
# Construct or look up the pcp group <A>.
A:=CreateIsomorphicPcGroup(pcgs,false,false);
UseFactorRelation( G, N, A );
# Construct the epimorphism from <G> onto <A>.
map := rec();
filter := IsPermGroupGeneralMappingByImages and
IsToPcGroupGeneralMappingByImages and
IsGroupGeneralMappingByPcgs and
IsMapping and IsSurjective and
HasSource and HasRange and
HasPreImagesRange and HasImagesSource and
HasKernelOfMultiplicativeGeneralMapping;
map.sourcePcgs := pcgs;
map.sourcePcgsImages := GeneratorsOfGroup( A );
ObjectifyWithAttributes( map,
NewType( GeneralMappingsFamily
( ElementsFamily( FamilyObj( G ) ),
ElementsFamily( FamilyObj( A ) ) ), filter ),
Source,G,
Range,A,
PreImagesRange,G,
ImagesSource,A,
KernelOfMultiplicativeGeneralMapping,N
);
return map;
end );
#############################################################################
##
#F PullBackNaturalHomomorphismsPool( <hom> )
##
InstallGlobalFunction(PullBackNaturalHomomorphismsPool,function(hom)
local s,r,nat,k;
s:=Source(hom);
r:=Range(hom);
for k in NaturalHomomorphismsPool(r).ker do
nat:=hom*NaturalHomomorphismByNormalSubgroup(r,k);
AddNaturalHomomorphismsPool(s,PreImage(hom,k),nat);
od;
end);
#############################################################################
##
#F TryQuotientsFromFactorSubgroups(<hom>,<ker>,<bound>)
##
InstallGlobalFunction(TryQuotientsFromFactorSubgroups,function(hom,ker,bound)
local s,p,k,it,u,v,d,ma,mak,lev,sub,low;
s:=Source(hom);
p:=Image(hom);
k:=KernelOfMultiplicativeGeneralMapping(hom);
it:=DescSubgroupIterator(p:skip:=4);
repeat
u:=NextIterator(it);
Info(InfoExtReps,2,"Factor subgroup index ",Index(p,u));
v:=PreImage(hom,u);
d:=DerivedSubgroup(v);
if not IsSubset(d,k) then
d:=ClosureGroup(ker,d);
if not IsSubset(d,k) then
ma:=NaturalHomomorphismByNormalSubgroup(v,d);
mak:=Image(ma,k);
lev:=0;
sub:=fail;
while sub=fail do
lev:=lev+1;
low:=ShallowCopy(LowLayerSubgroups(Range(ma),lev));
SortBy(low,x->-Size(x));
sub:=First(low,x->not IsSubset(x,mak));
od;
sub:=PreImage(ma,sub);
Info(InfoExtReps,2,"Found factor permrep ",IndexNC(s,sub));
d:=Core(s,sub);
AddNaturalHomomorphismsPool(s,d,sub);
k:=Intersection(k,d);
if Size(k)=Size(ker) then return;fi;
fi;
fi;
until IndexNC(p,u)>=bound;
end);
#############################################################################
##
#M UseFactorRelation( <num>, <den>, <fac> ) . . . . for perm group factors
##
InstallMethod( UseFactorRelation,
[ IsGroup and HasSize, IsObject, IsPermGroup ],
function( num, den, fac )
local limit;
if not HasSize( fac ) then
if HasSize(den) then
SetSize( fac, Size( num ) / Size( den ) );
else
limit := Size( num );
if IsBound( StabChainOptions(fac).limit ) then
limit := Minimum( limit, StabChainOptions(fac).limit );
fi;
StabChainOptions(fac).limit:=limit;
fi;
fi;
TryNextMethod();
end );
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