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fingolfin merged 2 commits into from
Sep 22, 2026
Merged

Refactor PrimitiveIdentification #101

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ca2efd7
Refactor PrimitiveIdentification
fingolfin Sep 21, 2026
1609926
Fix two errors in refactored PrimitiveIdentification
fingolfin Sep 21, 2026
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324 changes: 161 additions & 163 deletions lib/primitiv.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -758,221 +758,219 @@ local l,g,fac,mats,perms,v,t,filename,strm,r,dim,q,k;
end );

# local cache for `PrimitiveIdentification':
PRILD:=0;
PGICS:=[];
BindGlobal("PGIdCache", rec( deg := 0, data := [] ));

InstallMethod(PrimitiveIdentification,"generic",true,[IsPermGroup],0,
function(grp)
local dom,deg,PD,s,cand,a,p,s_quot,b,cs,n,beta,alpha,i,ag,bg,q,gl,hom,nr,c,x,conj;
local dom,deg,PD,s,cand,a,p,b,f,q,gl,hom;
dom:=MovedPoints(grp);
if not (IsTransitive(grp,dom) and IsPrimitive(grp,dom)) then
Error("Group must operate primitively");
fi;
deg:=Length(dom);
# through PRIMGrp, not PRIMGRP[deg]: an entry may be the call that produces
# it, and PD[i][2] below would be indexing a function. The upper range has
# always come this way; the lower one now does too.
PD:=List([1 .. NrPrimitiveGroups(deg)], t -> PRIMGrp(deg, t));

if IsNaturalAlternatingGroup(grp) then
SetSize(grp, Factorial(deg)/2);
elif IsNaturalSymmetricGroup(grp) then
SetSize(grp, Factorial(deg));
fi;

# size
s:=Size(grp);
cand:=Filtered([1..PRIMLENGTHS[deg]],i->PD[i][2]=s);

#ons
if Length(cand)>1 and Length(Set(PD{cand},i->i[4]))>1 then
a:=ONanScottType(grp);
cand:=Filtered(cand,i->PD[i][4]=a);
# treat alternating and symmetric group: these are uniquely determined by
# their size, so it is worth it to ensure we have it already
if not HasSize(grp) then
if IsNaturalAlternatingGroup(grp) then
SetSize(grp, Factorial(deg)/2);
elif IsNaturalSymmetricGroup(grp) then
SetSize(grp, Factorial(deg));
fi;
fi;

# suborbits
if Length(cand)>1 and Length(Set(PD{cand},i->i[5]))>1 then
s:=Stabilizer(grp,dom[1]);
a:=Collected(OrbitLengths(s,dom{[2..Length(dom)]}));
cand:=Filtered(cand,i->Set(PD[i][5])=Set(a));
# get information of all primitive groups of the given degree
PD := List([1 .. NrPrimitiveGroups(deg)], t -> PRIMGrp(deg, t));

#
# filter by size
#
s := Size(grp);
PD := Filtered(PD, primdata -> primdata[2] = s);
if Length(PD)=1 then return PD[1][1]; fi;

#
# filter by ONanScottType if it is not unique
#
if Length(Set(PD, primdata -> primdata[4])) > 1 then
a := ONanScottType(grp);
PD := Filtered(PD, primdata -> primdata[4] = a);
fi;

# Transitivity
if Length(cand)>1 and Length(Set(PD{cand},i->i[6]))>1 then
a:=Transitivity(grp,dom);
cand:=Filtered(cand,i->PD[i][6]=a);
if Length(PD)=1 then return PD[1][1]; fi;

#
# filter by suborbit lengths
#
if Length(Set(PD, primdata -> primdata[5])) > 1 then
s := Stabilizer(grp, dom[1]);
a := Set(Collected(OrbitLengths(s, dom{[2..Length(dom)]})));
PD := Filtered(PD, primdata -> Set(primdata[5]) = a);
fi;
if Length(PD)=1 then return PD[1][1]; fi;

#
# filter by transitivity if it is not unique
#
if Length(Set(PD, primdata -> primdata[6])) > 1 then
a := Transitivity(grp, dom);
PD := Filtered(PD, primdata -> primdata[6] = a);
fi;
if Length(PD)=1 then return PD[1][1]; fi;

if Length(cand)>1 then
# now we need to create the groups
p:=List(cand,i->PrimitiveGroup(deg,i));

# in product action case, some tests on the socle quotient.
if ONanScottType(grp) = "4c" then
#first we just identify its isomorphism type
s:= Socle(grp);
s_quot:= FactorGroup(grp, s);
a:= IdGroup(s_quot);
b:= [];
for i in [1..Length(cand)] do
b[i]:= IdGroup(FactorGroup(p[i], Socle(p[i])));
od;
s:= Filtered([1..Length(cand)], i->b[i] =a);
cand:= cand{s};
p:= p{s};
fi;
# now we need to create the groups
cand := List(PD, primdata -> PrimitiveGroup(deg, primdata[1]));

# in product action case, some tests on the socle quotient.
if ONanScottType(grp) = "4c" then
# first we just identify its isomorphism type
f := g -> IdGroup(FactorGroup(g, Socle(g)));
a := f(grp);
cand := Filtered(cand, g -> f(g) = a);
fi;
if Length(cand) = 1 then return PrimitiveIdentification(cand[1]); fi;

#
# AbelianInvariants
if Length(cand)>1 then
a:= AbelianInvariants(grp);
b:= [];
for i in [1..Length(cand)] do
b[i]:= AbelianInvariants(p[i]);
od;
s:= Filtered([1..Length(cand)], i->b[i] =a);
cand:= cand{s};
p:= p{s};
#
if Length(cand) > 1 then
a := AbelianInvariants(grp);
cand := Filtered(cand, g -> AbelianInvariants(g) = a);
fi;

if Length(cand)>1 then
#
# sylow orbits
#
if Length(cand) > 1 then
# sylow orbits
gl:=Reversed(PrimeDivisors(Size(grp)));
while Length(cand)>1 and Length(gl)>0 do
s:=SylowSubgroup(grp,gl[1]);
a:=Collected(OrbitLengths(s,MovedPoints(grp)));
b:=[];
for i in [1..Length(cand)] do
s:=SylowSubgroup(p[i],gl[1]);
b[i]:=Collected(OrbitLengths(s,MovedPoints(p[i])));
od;
s:=Filtered([1..Length(cand)],i->b[i]=a);
cand:=cand{s};
p:=p{s};
gl:=gl{[2..Length(gl)]};
gl := ShallowCopy(PrimeDivisors(Size(grp)));
f := function(g)
local s;
s := SylowSubgroup(g, p);
return Collected(OrbitLengths(s, MovedPoints(g)));
end;
while Length(cand) > 1 and Length(gl) > 0 do
p := Remove(gl); # largest remaining prime divisor
a := f(grp);
cand := Filtered(cand, g -> f(g) = a);
od;
fi;

#
# Some further tests for the sylow subgroups
#
if Length(cand) > 1 then
# Some further tests for the sylow subgroups
for q in PrimeDivisors(Size(grp)/Size(Socle(grp))) do
if q=1 then
q:=2;
fi;

ag:=Image(IsomorphismPcGroup(SylowSubgroup(grp,q)));
# central series
a:=List(LowerCentralSeries(ag),Size);
b:=[];
for i in [1..Length(cand)] do
bg:=Image(IsomorphismPcGroup(SylowSubgroup(p[i],q)));
b[i]:=List(LowerCentralSeries(bg),Size);
od;
s:=Filtered([1..Length(cand)],i->b[i]=a);
cand:=cand{s};
p:=p{s};

if Length(cand)>1 then
f := function(g)
local ag;
ag:=Image(IsomorphismPcGroup(SylowSubgroup(g,q)));
return List(LowerCentralSeries(ag), Size);
end;
a := f(grp);
cand := Filtered(cand, g -> f(g) = a);

if Length(cand) > 1 then
# Frattini subgroup
a:=Size(FrattiniSubgroup(ag));
b:=[];
for i in [1..Length(cand)] do
bg:=Image(IsomorphismPcGroup(SylowSubgroup(p[i],q)));
b[i]:=Size(FrattiniSubgroup(bg));
od;
s:=Filtered([1..Length(cand)],i->b[i]=a);
cand:=cand{s};
p:=p{s};
f := function(g)
local ag;
ag:=Image(IsomorphismPcGroup(SylowSubgroup(g,q)));
return Size(FrattiniSubgroup(ag));
end;
a := f(grp);
cand := Filtered(cand, g -> f(g) = a);
fi;

if Length(cand)>1 and Size(ag)<512 then
# Isomorphism type of 2-Sylow
a:=IdGroup(ag);
b:=[];
for i in [1..Length(cand)] do
bg:=Image(IsomorphismPcGroup(SylowSubgroup(p[i],q)));
b[i]:=IdGroup(bg);
od;
s:=Filtered([1..Length(cand)],i->b[i]=a);
cand:=cand{s};
p:=p{s};
if Length(cand) > 1 and Size(SylowSubgroup(grp,q)) < 512 then
# Isomorphism type of Sylow subgroup
f := function(g)
local ag;
ag:=Image(IsomorphismPcGroup(SylowSubgroup(g,q)));
return IdGroup(ag);
end;
a := f(grp);
cand := Filtered(cand, g -> f(g) = a);
fi;

od;
fi;

#back for a closer look at the product action groups.
# back for a closer look at the product action groups.
if Length(cand) > 1 and ONanScottType(grp) = "4c" then
#just here out of curiosity during testing.
#Print("cand =", cand, "\n");
#now we construct the action of the socle quotient as a
#(necessarily transitive) action on the socle factors.
s:= Socle(grp);
cs:= CompositionSeries(s);
cs:= cs[Length(cs)-1];
n:= Normalizer(grp, cs);
beta:= FactorCosetAction(grp, n);
alpha:= FactorCosetAction(n, ClosureGroup(Centralizer(n, cs), s));
a:= TransitiveIdentification(Group(KuKGenerators(grp, beta, alpha)));
b:= [];
for i in [1..Length(cand)] do
s:= Socle(p[i]);
cs:= CompositionSeries(s);
cs:= cs[Length(cs)-1];
n:= Normalizer(p[i], cs);
beta:= FactorCosetAction(p[i], n);
alpha:= FactorCosetAction(n, ClosureGroup(Centralizer(n, cs), s));
b[i]:= TransitiveIdentification(Group(KuKGenerators(p[i], beta, alpha)));
od;
s:= Filtered([1..Length(cand)], i->b[i]=a);
cand:= cand{s};
p:= p{s};
# construct the action of the socle quotient as a (necessarily transitive)
# action on the socle factors.
f := function(g)
local s, cs, n, beta, alpha;
s := Socle(g);
cs := CompositionSeries(s);
cs := cs[Length(cs)-1];
n := Normalizer(g, cs);
beta := FactorCosetAction(g, n);
alpha := FactorCosetAction(n, ClosureGroup(Centralizer(n, cs), s));
return TransitiveIdentification(Group(KuKGenerators(g, beta, alpha)));
end;
a := f(grp);
cand := Filtered(cand, g -> f(g) = a);
fi;

if Length(cand)>1 then
# Klassen
a:=Collected(List(ConjugacyClasses(grp:onlysizes),
i->[CycleStructurePerm(Representative(i)),Size(i)]));
#
# conjugacy classes
#
if Length(cand) > 1 then

# use caching
if deg<>PRILD then
PRILD:=deg;
PGICS:=[];
if deg <> PGIdCache.deg then
PGIdCache.deg := deg;
PGIdCache.data := [];
fi;

b:=[];
for i in [1..Length(cand)] do
if not IsBound(PGICS[cand[i]]) then
PGICS[cand[i]]:=Collected(List(ConjugacyClasses(p[i]:onlysizes),
j->[CycleStructurePerm(Representative(j)),Size(j)]));
# compute the fingerprint, with cache lookup
f := function(g)
local i, dat;
if HasPrimitiveIdentification(g) then
# is it in the cache?
i := PrimitiveIdentification(g);
if IsBound(PGIdCache.data[i]) then
return PGIdCache.data[i];
fi;
fi;
b[i]:=PGICS[cand[i]];
od;
dat := ConjugacyClasses(g : onlysizes);
dat := List(dat, i -> [CycleStructurePerm(Representative(i)), Size(i)]);
dat := Collected(dat);
if IsBound(i) then
PGIdCache.data[i] := dat;
fi;
return dat;
end;

s:=Filtered([1..Length(cand)],i->b[i]=a);
cand:=cand{s};
p:=p{s};
a := f(grp);
cand := Filtered(cand, g -> f(g) = a);
fi;

if Length(cand)>1 and ForAll(p,i->ONanScottType(i)="1")
and ONanScottType(grp)="1" then
gl:=Factors(NrMovedPoints(grp));
gl:=GL(Length(gl),gl[1]);
hom:=IsomorphismPermGroup(gl);
s:=List(p,i->Subgroup(gl,LinearActionLayer(i,Pcgs(Socle(i)))));
b:=Subgroup(gl,LinearActionLayer(grp,Pcgs(Socle(grp))));
s:=Filtered([1..Length(cand)],
i->RepresentativeAction(Image(hom,gl),Image(hom,s[i]),Image(hom,b))<>fail);
cand:=cand{s};
p:=p{s};
#
# affine case
#
if Length(cand) > 1 and ONanScottType(grp) = "1" then
Assert(0, ForAll(cand, g -> ONanScottType(g) = "1"));

# degree must be a prime power
gl := Factors(deg);
gl := GL(Length(gl), gl[1]);
hom := IsomorphismPermGroup(gl); # nasty

# map group by its linear action into GL, and then further via hom
f := g -> Image(hom, Subgroup(gl, LinearActionLayer(g, Pcgs(Socle(g)))));
a := f(grp);
b := Image(hom, gl);
cand := Filtered(cand, g -> RepresentativeAction(b, f(g), a) <> fail);
fi;

if Length(cand)=1 then
return cand[1];
else
if Length(cand) > 1 then
Error("Uh-Oh, this should never happen ",cand);
return cand[1];
fi;
return PrimitiveIdentification(cand[1]);
end);

InstallMethod(SimsNo,"via `PrimitiveIdentification'",true,[IsPermGroup],0,
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