Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place. Commercial Alternative to JupyterHub.
Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place. Commercial Alternative to JupyterHub.
| Download
GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 521449###########################################################
## CatOneGroupByCrossedModule
## XmodToHAP
## CatOneGroupsByGroup
## HomotopyCatOneGroup
## NumberSmallCatOneGroups
## SmallCatOneGroup
## IdCatOneGroup
############################################################
#############################################################################
#0
#F CatOneGroupByCrossedModule
## Input: Either a crossed module, or a morphism of crossed modules, or
## a sequence of morphisms of crossed modules
## Output: The image of the input under the functor
## (crossed modules)->(cat-1-groups).
##
InstallGlobalFunction(CatOneGroupByCrossedModule, function(X)
local
CMToCat1_Obj,
CMToCat1_Morpre,
CMToCat1_Mor,
CMToCat1_Seq;
######################################################################
#1
## CMToCat1_Obj
## Input: A crossed module X
## Output: The cat-1-group corresponds to X
##
CMToCat1_Obj:=function(XC)
local
d,act,p,m,M,P,AutM,GensM,GensP,alpha,
G,GensG,pro,emb1,emb2,Elts,Eltt,g,pg,s,t;
d:=XC!.map;
act:=XC!.action;
M:=Source(d);
P:=Range(d);
AutM:=AutomorphismGroup(M);
GensM:=GeneratorsOfGroup(M);
GensP:=GeneratorsOfGroup(P);
alpha:=GroupHomomorphismByImages(P,AutM,GensP,List(GensP,p->
GroupHomomorphismByImages(M,M,GensM,List(GensM,m->act(p,m)))));
G:=SemidirectProduct(P,alpha,M);
GensG:=GeneratorsOfGroup(G);
pro:=Projection(G);
emb1:=Embedding(G,1);
emb2:=Embedding(G,2);
Elts:=[];
Eltt:=[];
for g in GensG do
p:=Image(pro,g);
pg:=Image(emb1,p);
Add(Elts,pg);
m:=PreImagesRepresentative(emb2,pg^(-1)*g);
Add(Eltt,Image(emb1,p*Image(d,m)));
od;
s:=GroupHomomorphismByImages(G,G,GensG,Elts);
t:=GroupHomomorphismByImages(G,G,GensG,Eltt);
return Objectify(HapCatOneGroup,
rec(sourceMap:=s,
targetMap:=t
));
end;
##
############### end of CMToCat1_Obj ##################################
######################################################################
#1
## CMToCat1_Morpre
##
CMToCat1_Morpre:=function(CC,CD,map)
local
GC,GensGC,proC,emb1C,emb2C,g,
GD,fM,fP,p,m,emb1D,emb2D,ImGensGC;
GC:=Source(CC!.sourceMap);
GensGC:=GeneratorsOfGroup(GC);
GD:=Source(CD!.sourceMap);
fM:=map(1);
fP:=map(2);
proC:=Projection(GC);
emb1C:=Embedding(GC,1);
emb2C:=Embedding(GC,2);
emb1D:=Embedding(GD,1);
emb2D:=Embedding(GD,2);
ImGensGC:=[];
for g in GensGC do
p:=Image(proC,g);
m:=PreImagesRepresentative(emb2C,(Image(emb1C,p))^(-1)*g);
Add(ImGensGC,Image(emb1D,Image(fP,p))*Image(emb2D,Image(fM,m)));
od;
return Objectify(HapCatOneGroupMorphism,
rec(
source:= CC,
target:= CD,
mapping:= GroupHomomorphismByImages(GC,
GD,GensGC,ImGensGC)
));
end;
##
############### end of CMToCat1_Morpre ###############################
######################################################################
#1
## CMToCat1_Mor
## Input: A morphism fX of crossed modules
## Output: The morphism of cat-1-groups corresponds to fX
##
CMToCat1_Mor:=function(fX)
local CC,CD,map;
CC:=CMToCat1_Obj(fX!.source);
CD:=CMToCat1_Obj(fX!.target);
map:=fX!.mapping;
return CMToCat1_Morpre(CC,CD,map);
end;
##
############### end of CMToCat1_Mor ##################################
######################################################################
#1
## CMToCat1_Seq
## Input: A sequence LfX of morphisms of crossed modules LfX
## Output: The sequence of morphisms of cat-1-groups corresponds to fX
##
CMToCat1_Seq:=function(LfX)
local n,i,GC,Res;
n:=Length(LfX);
GC:=[];
for i in [1..n] do
GC[i]:=CMToCat1_Obj(LfX[i]!.source);
od;
GC[n+1]:=CMToCat1_Obj(LfX[i]!.target);
Res:=[];
for i in [1..n] do
Res[i]:=CMToCat1_Morpre(GC[i],GC[i+1],LfX[i]!.mapping);
od;
return Res;
end;
##
############### end of CMToCat1_Seq ##################################
if IsHapCrossedModule(X) then
return CMToCat1_Obj(X);
fi;
if IsHapCrossedModuleMorphism(X) then
return CMToCat1_Mor(X);
fi;
if IsList(X) then
return CMToCat1_Seq(X);
fi;
end);
##
#################### end of CatOneGroupByCrossedModule ######################
#############################################################################
#0
#F XmodToHAP
## Input: A cat-1-group from the Xmod package
## Output: The cat-1-group with data type from the HAP package
##
InstallGlobalFunction(XmodToHAP,function (X)
local G,Gens,s,t;
if IsHapCatOneGroup(X) then
return X;
fi;
if IsComponentObjectRep(X) then
if "TailMap" in NamesOfComponents(X) and
"HeadMap" in NamesOfComponents(X) then
G:=X!.Source;
Gens:=GeneratorsOfGroup(G);
s:=X!.TailMap;
t:=X!.HeadMap;
return Objectify(HapCatOneGroup,rec(
sourceMap:=GroupHomomorphismByImagesNC(G,G,Gens,List(Gens,
g->Image(s,g))),
targetMap:=GroupHomomorphismByImagesNC(G,G,Gens,List(Gens,
g->Image(t,g)))
));
fi;
fi;
Print("Argument is not a cat-1-group from the Xmod package.\n");
return fail;
end);
##
#################### end of XmodToHAP #######################################
#############################################################################
#0
#F CatOneGroupsByGroup
## Input: A group G
## Output: The list of all non-isomorphic cat-1-groups with underlying group G
##
InstallGlobalFunction(CatOneGroupsByGroup, function(G)
local
nk,n,k,Lst,S,p,x,tmp,i,Imgs,s,h,hinv,Gens,C,ResCats,
ClassifyPairsByOrbit,CreatePairsByAbelianGroup,
CreatePairsByNonAbelianGroup,CreatePairsByGroup;
######################################################################
#1
#F ClassifyPairsByOrbit
## Input: A list L of pairs of group homomorphisms [s,t]:G->G
## and the automorphism group of G.
## Output: A list of non-isomorphic pairs of L
##
ClassifyPairsByOrbit:=function(A,Lst)
local
CL,TmpCL,Lx,Res,
ActToMap,ActToPair,
RefineClassesUnderGroup,
processDuplicates;
if Length(Lst)<=1 then
return Lst;
fi;
#############################################################
#2
ActToMap:=function(s,f)
return InverseGeneralMapping(f)*s*f;
end;
##
#############################################################
#2
ActToPair:=function(p,f)
local h;
h:=InverseGeneralMapping(f);
return [h*p[1]*f,h*p[2]*f];
end;
##
#############################################################
#############################################################
#2
#F RefineClassesUnderGroup
##
RefineClassesUnderGroup:=function(A,Lst,Indx,attr,actAttr)
local
ValAttr,i,NC,LC,Sel,Orb,T,Dict,
S,Gens,op,qs,g,h,img,p,x,cnt;
ValAttr:=[];
for i in [1..Length(Indx)] do
ValAttr[i]:=attr(Lst[Indx[i]]);
od;
NC:=[];
Gens:=SmallGeneratingSet(A);
Sel:=[1..Length(Indx)];
while Length(Sel)>0 do
# orbit algorithm on attributes, regular transversal
LC:=[Indx[Sel[1]]];
Orb:=[ValAttr[Sel[1]]];
Unbind(Sel[1]);
T:=[One(A)];
Dict:=NewDictionary(Orb[1],true);
AddDictionary(Dict,Orb[1],1);
S:=TrivialSubgroup(A);
op:=1;
qs:=Size(A);
while op<=Length(Orb) and Size(S)<qs do
for g in Gens do
img:=actAttr(Orb[op],g);
p:=LookupDictionary(Dict,img);
if p=fail then
Add(Orb,img);
AddDictionary(Dict,img,Length(Orb));
Add(T,T[op]*g);
qs:=Size(A)/Length(Orb);
elif Size(S)<=qs/2 then
x:=T[op]*g/T[p];
S:=ClosureSubgroup(S,x);
fi;
od;
op:=op+1;
od;
# which other values are in the orbit
for i in [2..Length(Sel)] do
p:=LookupDictionary(Dict,ValAttr[Sel[i]]);
if p<>fail then
x:=Indx[Sel[i]];
AddSet(LC,x);
Unbind(Sel[i]);
if p>1 then # not identity
h:=T[p]^-1;
Lst[x]:=ActToPair(Lst[x],h);
fi;
fi;
od;
Add(NC,[S,LC]);
Sel:=Set(Sel);
od;
return NC;
end;
##
########## end of RefineClassesUnderGroup ###################
#############################################################
#2
#P processDuplicates #remove duplicates
##
processDuplicates:=function()
local Lx,i,p,Sel;
TmpCL:=[];
for Lx in CL do
Sel:=[];
for i in Lx[2] do
p:=First(Sel,x->Lst[i]=Lst[x]);
if p=fail then
Add(Sel,i);
fi;
od;
Add(TmpCL,[Lx[1],Sel]);
od;
CL:=TmpCL;
end;
##
########## end of processDuplicates #########################
############### Images of first component ###################
CL:=RefineClassesUnderGroup(A,Lst,[1..Length(Lst)],x->Image(x[1]),
function(s,a) return Image(a,s);end);
processDuplicates();
Res:=[];
############### Kernels of first component ##################
TmpCL:=[];
for Lx in CL do
if Length(Lx[2])= 1 then
Add(Res,Lst[Lx[2][1]]);
else
Append(TmpCL,RefineClassesUnderGroup(Lx[1],Lst,Lx[2],
x->Kernel(x[1]),function(s,a) return Image(a,s);end));
fi;
od;
CL:=TmpCL;
processDuplicates();
############### Kernels of second component #################
TmpCL:=[];
for Lx in CL do
if Length(Lx[2])= 1 then
Add(Res,Lst[Lx[2][1]]);
else
Append(TmpCL,RefineClassesUnderGroup(Lx[1],Lst,Lx[2],
x->Kernel(x[2]),function(s,a) return Image(a,s);end));
fi;
od;
CL:=TmpCL;
processDuplicates();
############### First component #############################
TmpCL:=[];
for Lx in CL do
if Length(Lx[2])= 1 then
Add(Res,Lst[Lx[2][1]]);
else
Append(TmpCL,RefineClassesUnderGroup(Lx[1],Lst,Lx[2],
x->x[1],ActToMap));
fi;
od;
CL:=TmpCL;
processDuplicates();
############### Second component ############################
TmpCL:=[];
for Lx in CL do
if Length(Lx[2])= 1 then
Add(Res,Lst[Lx[2][1]]);
else
Append(TmpCL,RefineClassesUnderGroup(Lx[1],Lst,Lx[2],
x->x[2],ActToMap));
fi;
od;
CL:=TmpCL;
processDuplicates();
if IsEmpty(CL) then
return Res;
fi;
if ForAny(CL,x->Length(x[2])>1) then
Error("Uniqueness failure");
fi;
return Concatenation(Res,List(CL,x->Lst[x[2][1]]));
end;
##
################ end of ClassifyPairsByOrbit #########################
######################################################################
#1
#F CreatePairsByAbelianGroup
## Input: An abelian group G
## Output: A list of non-isomorphic pairs [s,t]:G->G such that (G,s,t)
## is a cat-1-group
##
CreatePairsByAbelianGroup:=function(G)
local
AbIn,NumX,SizeX,nX,GroupX,GensX,
i,LSX,e,Gens,sum,
GensLK,LK,sLK,LComX,tmp,GensK,Imgs,xK,xG,
M,S,f,finv,nLK,Aut,n,LfK,K,CompK,N,GensN,t,
ResPairs,FCombination;
AbIn:=AbelianInvariants(G);
if IsEmpty(AbIn) then
return [IdentityMapping(G),IdentityMapping(G)];
fi;
NumX:=Set(AbIn);
SizeX:=List(NumX,x->Length(Filtered(AbIn,i->i=x)));
#################################################################
##
FCombination:=function(n)
local
T,ST,tmp,Res,x;
if n=1 then
return List([0..SizeX[n]],x->[x]);
fi;
if n>1 then
Res:=[];
T:=FCombination(n-1);
for x in [0..SizeX[n]] do
ST:=StructuralCopy(T);
for tmp in ST do
Add(tmp,x);
od;
Append(Res,ST);
od;
return Res;
fi;
end;
##
#################################################################
nX:=Length(NumX);
GroupX:=[];
GensX:=[];
for i in [1..nX] do
GroupX[i]:=CyclicGroup(NumX[i]);
GensX[i]:=First(GroupX[i],g->Order(g)=NumX[i]);
od;
LSX:=[];
for i in [1..nX] do
Append(LSX,List([1..SizeX[i]],m->GroupX[i]));
od;
S:=DirectProduct(LSX);
e:=One(S);
Gens:=[];
sum:=0;
for i in [1..nX] do
Append(Gens,List([1..SizeX[i]],m->Image(Embedding(S,sum+m),GensX[i])));
sum:=sum+SizeX[i];
od;
GensLK:=[];
LK:=[];
sLK:=[];
LComX:=FCombination(Length(NumX));
for tmp in LComX do
GensK:=[];
Imgs:=[];
sum:=0;
for i in [1..Length(tmp)] do
xK:=List([1..tmp[i]],m->Gens[sum+m]);
xG:=Concatenation(xK,List([tmp[i]+1..SizeX[i]],m->e));
Append(GensK,xK);
Append(Imgs,xG);
sum:=sum+SizeX[i];
od;
Add(GensLK,GensK);
if IsEmpty(GensK) then
Add(LK,Group(e));
else
Add(LK,Group(GensK));
fi;
Add(sLK,GroupHomomorphismByImages(S,S,Gens,Imgs));
od;
f:=IsomorphismGroups(S,G);
finv:=InverseGeneralMapping(f);
LK:=List(LK,K->Image(f,K));
sLK:=List(sLK,s->finv*s*f);
GensLK:=List(LK,K->GeneratorsOfGroup(K));
nLK:=Length(LK);
Aut:=AutomorphismGroup(G);
e:=One(G);
ResPairs:=[];
for n in [1..nLK] do
LfK:=[];
K:=LK[n];
CompK:=Complementclasses(G,K);
for N in CompK do
GensN:=SmallGeneratingSet(N);
Gens:=Concatenation(GensLK[n],GensN);
Imgs:=Concatenation(GensLK[n],List(GensN,g->e));
t:=GroupHomomorphismByImages(G,G,Gens,Imgs);
Add(LfK,[sLK[n],t]);
od;
Append(ResPairs,ClassifyPairsByOrbit(Aut,LfK));
od;
return ResPairs;
end;
##
############### end of CreatePairsByAbelianGroup #####################
######################################################################
#1
#F CreatePairsByNonAbelianGroup
## Input: An non-abelian group G
## Output: A list of non-isomorphic pairs [s,t]:G->G such that (G,s,t)
## is a cat-1-group
##
CreatePairsByNonAbelianGroup:=function(G)
local
Aut,GensG,LN,nLN,IdLN,SizeLN,SetSizeLN,CLN,
i,j,k,nCLN,LS,IdLS,nLS,
n,L,nL,LfN,dem,N,nat,GoN,SizeGoN,IdGoN,K,GensK,f,h,
x,y,s,Li,SKer,NotSKer,a,b,
T,PairSKer,PairNotSKer,ResPairs;
Aut:=AutomorphismGroup(G);
GensG:=GeneratorsOfGroup(G);
LN:=NormalSubgroups(G);
nLN:=Length(LN);
IdLN:=List(LN,x->IdGroup(x));
SizeLN:=List(LN,x->Size(x));
SetSizeLN:=Set(SizeLN);
CLN:=List([1..Length(SetSizeLN)],x->[]);
for i in [1..nLN] do
Add(CLN[Position(SetSizeLN,SizeLN[i])],i);
od;
nCLN:=Length(CLN);
LS:=LatticeSubgroups(G)!.conjugacyClassesSubgroups;
if not IsMutable(LS) then
LS:= ShallowCopy(LS);
fi;
LS:=List(LS,x->x[1]);
IdLS:=List(LS,x->IdGroup(x));
nLS:=Length(LS);
PairSKer:=[];
PairNotSKer:=[];
for n in [1..nCLN] do
L:=CLN[n];
nL:=Length(L);
LfN:=List([1..nLN],x->[]);
for i in L do
N:=LN[i];
nat:=NaturalHomomorphismByNormalSubgroup(G,N);
GoN:=Range(nat);
SizeGoN:=Size(GoN);
IdGoN:=IdGroup(GoN);
for k in [1..nLS] do
if IdLS[k]= IdGoN then
K:=LS[k];
if Size(Image(nat,K))=SizeGoN then
GensK:=GeneratorsOfGroup(K);
f:=GroupHomomorphismByImages(K,GoN,GensK,
List(GensK,g->Image(nat,g)));
h:=InverseGeneralMapping(f);
s:=GroupHomomorphismByImages(G,G,GensG,
List(GensG,g->Image(h,Image(nat,g))));
Add(LfN[i],[k,s]);
fi;
fi;
od;
od;
SKer:=[];
for a in [1..nL] do
i:=L[a];
Li:=[];
if Size(CommutatorSubgroup(LN[i],LN[i]))=1 then
Li:=List(LfN[i],x->[x[2],x[2]]);
fi;
Append(SKer,ClassifyPairsByOrbit(Aut,Li));
od;
Append(PairSKer,ClassifyPairsByOrbit(Aut,SKer));
NotSKer:=[];
for a in [2..nL] do
i:=L[a];
Li:=[];
for b in [1..a-1] do
j:=L[b];
if Size(CommutatorSubgroup(LN[i],LN[j]))=1 then
for x in LfN[i] do
for y in LfN[j] do
if x[1]=y[1] then
Add(Li,[x[2],y[2]]);
fi;
od;od;
fi;
od;
Append(NotSKer,ClassifyPairsByOrbit(Aut,Li));
od;
NotSKer:=ClassifyPairsByOrbit(Aut,NotSKer);
T:=ShallowCopy(NotSKer);
for x in NotSKer do
Add(T,[x[2],x[1]]);
od;
Append(PairNotSKer,ClassifyPairsByOrbit(Aut,T));
od;
ResPairs:=Concatenation(PairSKer,PairNotSKer);
return ResPairs;
end;
##
############### end CreatePairsByNonAbelianGroup #####################
######################################################################
#1
CreatePairsByGroup:=function(G)
if IsAbelian(G) then
return CreatePairsByAbelianGroup(G);
else
return CreatePairsByNonAbelianGroup(G);
fi;
end;
##
######################################################################
n:=Size(G);
if n<=CATONEGROUP_DATA_SIZE then
nk:=IdGroup(G);
n:=nk[1];
k:=nk[2];
Gens:=GeneratorsOfGroup(G);
S:=SmallGroup(n,k);
h:=IsomorphismGroups(G,S);
hinv:=InverseGeneralMapping(h);
Lst:=[];
if nk in CATONEGROUP_DATA_PERM then
for x in CATONEGROUP_DATA[n][k] do
tmp:=[];
for i in [1..2] do
s:=GroupHomomorphismByImages(S,S,x[i][1],x[i][2]);
Imgs:=List(Gens,g->Image(hinv,Image(s,(Image(h,g)))));
tmp[i]:=GroupHomomorphismByImages(G,G,Gens,Imgs);
od;
Add(Lst,tmp);
od;
else
p:=Pcgs(S);
for x in CATONEGROUP_DATA[n][k] do
tmp:=[];
for i in [1..2] do
Imgs:=List(x[i],m->PcElementByExponents(p,m));
s:=GroupHomomorphismByImages(S,S,p,Imgs);
Imgs:=List(Gens,g->Image(hinv,Image(s,(Image(h,g)))));
tmp[i]:=GroupHomomorphismByImages(G,G,Gens,Imgs);
od;
Add(Lst,tmp);
od;
fi;
else
Lst:=CreatePairsByGroup(G);
fi;
ResCats:=[];
for x in Lst do
C:=Objectify(HapCatOneGroup,
rec(sourceMap:=x[1],
targetMap:=x[2]
));
Add(ResCats,C);
od;
return ResCats;
end);
##
################### end of CatOneGroupsByGroup ##############################
#############################################################################
#0
#F HomotopyCatOneGroup
## Input:
## Output:
##
InstallGlobalFunction(HomotopyCatOneGroup, function(C)
if Order(HomotopyGroup(C,1))*Order(HomotopyGroup(C,2))=Order(C) then
return C;
fi;
return CatOneGroupByCrossedModule(HomotopyCrossedModule(
CrossedModuleByCatOneGroup(C)));
end);
##
################### end of NumberSmallCatOneGroups ##########################
#############################################################################
#0
#F NumberSmallCatOneGroups
## Input: Integers n,k
## Output: The number of non-isomorphic cat-1-groups of SmallGroup(n,k)
##
InstallGlobalFunction(NumberSmallCatOneGroups, function(arg)
local n,k;
n:=arg[1];
if n >CATONEGROUP_DATA_SIZE then
Print("This function only apply for order less than or equal to ",
CATONEGROUP_DATA_SIZE,".\n");
return fail;
fi;
if Length(arg)=1 then
return Sum(List([1..NumberSmallGroups(n)],k->Length(CATONEGROUP_DATA[n][k])));
fi;
if Length(arg)=2 then
k:=arg[2];
if k>NumberSmallGroups(n) then
Print("There are only ",NumberSmallGroups(n)," groups of order ",n,"\n");
return fail;
fi;
return Length(CATONEGROUP_DATA[n][k]);
fi;
return fail;
end);
##
################### end of NumberSmallCatOneGroups ##########################
#############################################################################
#0
#F SmallCatOneGroup
## Input: Integer numbers n,k,i.
## Output: The ith cat-1-group of the list of all non-isomorphic cat-1-groups
## of underlying group SmallGroup(n,k).
##
InstallGlobalFunction(SmallCatOneGroup, function(arg)
local S,m,sum,x,s,t,p,
n,k,i;
n:=arg[1];
k:=arg[2];
if n > CATONEGROUP_DATA_SIZE then
Print("This function only apply for order less than or equal to ",
CATONEGROUP_DATA_SIZE,".\n");
return fail;
fi;
if Length(arg)=2 then
m:=NumberSmallCatOneGroups(n);
if k>m then
Print("There are only ",m," cat-1-groups of order ",n,"\n");
return fail;
fi;
sum:=0;
m:=0;
while sum<k do
i:=k-sum;
m:=m+1;
sum:=sum+Length(CATONEGROUP_DATA[n][m]);
od;
k:=m;
fi;
if Length(arg)=3 then
i:=arg[3];
m:=Length(CATONEGROUP_DATA[n][k]);
if i>m then
Print("There are only ",m," cat-1-groups of SmallGroup(",n,",",k,")\n");
return fail;
fi;
fi;
S:=SmallGroup(n,k);
x:=CATONEGROUP_DATA[n][k][i];
if [n,k] in CATONEGROUP_DATA_PERM then
s:=GroupHomomorphismByImages(S,S,x[1][1],x[1][2]);
t:=GroupHomomorphismByImages(S,S,x[2][1],x[2][2]);
else
p:=Pcgs(S);
s:=GroupHomomorphismByImages(S,S,p,List(x[1],
m->PcElementByExponents(p,m)));
t:=GroupHomomorphismByImages(S,S,p,List(x[2],
m->PcElementByExponents(p,m)));
fi;
return Objectify(HapCatOneGroup,
rec(sourceMap:=s,
targetMap:=t
));
end);
##
################### end of SmallCatOneGroup #################################
#############################################################################
#0
#F IdCatOneGroup
##
InstallGlobalFunction(IdCatOneGroup, function(C)
local
ActToMap,ActToPair,ActToSubgroup,FindOrbit,processOrbit,
s,t,G,nk,n,k,S,f,Lst,x,tmp,i,p,Imgs,A,xC,Ln,CLn,attr,actAttr,M;
######################################################################
#1
ActToMap:=function(s,f)
return InverseGeneralMapping(f)*s*f;
end;
##
######################################################################
#1
ActToPair:=function(p,f)
local h;
h:=InverseGeneralMapping(f);
return [h*p[1]*f,h*p[2]*f];
end;
##
######################################################################
#1
ActToSubgroup:=function(K,f)
return Image(f,K);
end;
##
#######################################################################
######################################################################
#1
#F FindOrbit
##
FindOrbit:=function(A,attr,actAttr)
local
Gens,Orb,T,Dict,S,op,qs,g,p,img,h;
Gens:=SmallGeneratingSet(A);
Orb:=[attr(xC)];
T:=[One(A)];
Dict:=NewDictionary(Orb[1],true);
AddDictionary(Dict,Orb[1],1);
S:=TrivialSubgroup(A);
op:=1;
qs:=Size(A);
while op<=Length(Orb) and Size(S)<qs do
for g in Gens do
img:=actAttr(Orb[op],g);
p:=LookupDictionary(Dict,img);
if p=fail then
Add(Orb,img);
AddDictionary(Dict,img,Length(Orb));
Add(T,T[op]*g);
qs:=Size(A)/Length(Orb);
elif Size(S)<=qs/2 then # otherwise stabilizer cant grow
h:=T[op]*g/T[p];
S:=ClosureSubgroup(S,h);
fi;
od;
op:=op+1;
od;
return [Dict,S,T];
end;
##
######################################################################
##
processOrbit:=function()
M:=FindOrbit(A,attr,actAttr);
for i in Ln do
p:=LookupDictionary(M[1],attr(Lst[i]));
if p<>fail then
Add(CLn,i);
Lst[i]:=ActToPair(Lst[i],M[3][p]^-1);
fi;
od;
end;
##
######################################################################
s:= C!.sourceMap;
t:= C!.targetMap;
G:=Source(s);
n:=Size(G);
if n>CATONEGROUP_DATA_SIZE then
Print("This function only apply for cat-1-groups of order less than ",
CATONEGROUP_DATA_SIZE+1,"\n");
return fail;
fi;
nk:=IdGroup(G);
k:=nk[2];
S:=SmallGroup(n,k);
f:=IsomorphismGroups(G,S);
Lst:=[];
if nk in CATONEGROUP_DATA_PERM then
for x in CATONEGROUP_DATA[n][k] do
tmp:=[];
for i in [1..2] do
tmp[i]:=GroupHomomorphismByImages(S,S,x[i][1],x[i][2]);
od;
Add(Lst,tmp);
od;
else
p:=Pcgs(S);
for x in CATONEGROUP_DATA[n][k] do
tmp:=[];
for i in [1..2] do
Imgs:=List(x[i],m->PcElementByExponents(p,m));
tmp[i]:=GroupHomomorphismByImages(S,S,p,Imgs);
od;
Add(Lst,tmp);
od;
fi;
A:=AutomorphismGroup(S);
xC:=ActToPair([s,t],f);
############## Image of first component #########################
A:=AutomorphismGroup(S);
Ln:=[1..Length(Lst)];
CLn:=[];
attr:=s->Image(s[1]);
actAttr:=ActToSubgroup;
processOrbit();
if Length(CLn) =1 then
return [n,k,CLn[1]];
fi;
A:=M[2];
Ln:=CLn;
CLn:=[];
############## Kernel of first component ########################
attr:=s->Kernel(s[1]);
actAttr:=ActToSubgroup;
processOrbit();
if Length(CLn) =1 then
return [n,k,CLn[1]];
fi;
A:=M[2];
Ln:=CLn;
CLn:=[];
############## Image of second component ########################
attr:=s->Image(s[2]);
actAttr:=ActToSubgroup;
processOrbit();
if Length(CLn) =1 then
return [n,k,CLn[1]];
fi;
A:=M[2];
Ln:=CLn;
CLn:=[];
############## Kernel of second component #######################
attr:=s->Kernel(s[1]);
actAttr:=ActToSubgroup;
processOrbit();
if Length(CLn) =1 then
return [n,k,CLn[1]];
fi;
A:=M[2];
Ln:=CLn;
CLn:=[];
############## First component ##################################
attr:=s->s[1];
actAttr:=ActToMap;
processOrbit();
if Length(CLn) =1 then
return [n,k,CLn[1]];
fi;
A:=M[2];
Ln:=CLn;
CLn:=[];
############## Second component #################################
attr:=s->s[2];
actAttr:=ActToMap;
processOrbit();
if Length(CLn) =1 then
return [n,k,CLn[1]];
fi;
return fail;
end);
##
##################### end of IdCatOneGroup ##################################
#############################################################################
#0
#F IsomorphismCatOneGroups
## Input: Two cat-1-groups C and D
## Output: A morphism between C and D if C and D are isomorphic.
## return false for other else
##
InstallGlobalFunction(IsomorphismCatOneGroups, function(C,D)
local
sC,tC,G,sD,tD,GD,
xC,xD,A,attr,actAttr,M,p,f,h,
ActToMap,ActToPair,ActToSubgroup,FindOrbit,processOrbit,
Map,map;
######################################################################
#1
ActToMap:=function(s,f)
return InverseGeneralMapping(f)*s*f;
end;
##
######################################################################
#1
ActToPair:=function(p,f)
local h;
h:=InverseGeneralMapping(f);
return [h*p[1]*f,h*p[2]*f];
end;
##
######################################################################
#1
ActToSubgroup:=function(K,f)
return Image(f,K);
end;
##
######################################################################
#1
#F FindOrbit
##
FindOrbit:=function(A,attr,actAttr)
local
Gens,Orb,T,Dict,S,op,qs,g,p,img,h;
Gens:=SmallGeneratingSet(A);
Orb:=[attr(xC)];
T:=[One(A)];
Dict:=NewDictionary(Orb[1],true);
AddDictionary(Dict,Orb[1],1);
S:=TrivialSubgroup(A);
op:=1;
qs:=Size(A);
while op<=Length(Orb) and Size(S)<qs do
for g in Gens do
img:=actAttr(Orb[op],g);
p:=LookupDictionary(Dict,img);
if p=fail then
Add(Orb,img);
AddDictionary(Dict,img,Length(Orb));
Add(T,T[op]*g);
qs:=Size(A)/Length(Orb);
elif Size(S)<=qs/2 then # otherwise stabilizer cant grow
h:=T[op]*g/T[p];
S:=ClosureSubgroup(S,h);
fi;
od;
op:=op+1;
od;
return [Dict,S,T];
end;
##
######################################################################
##
processOrbit:=function()
M:=FindOrbit(A,attr,actAttr);
p:=LookupDictionary(M[1],attr(xD));
if p=fail then
return fail;
fi;
A:=M[2];
if p<>1 then
h:=M[3][p]^-1;
f:=f*h;
xD:=ActToPair(xD,h);
fi;
end;
##
######################################################################
######################################################################
sC:=C!.sourceMap;
tC:=C!.targetMap;
G:=Source(sC);
sD:=D!.sourceMap;
tD:=D!.targetMap;
GD:=Source(sD);
f:=IsomorphismGroups(GD,G);
if f = fail then
return fail;
fi;
if IsomorphismGroups(HomotopyGroup(C,1),HomotopyGroup(D,1))=fail then
return fail;
fi;
if IsomorphismGroups(HomotopyGroup(C,2),HomotopyGroup(D,2))=fail then
return fail;
fi;
if IsomorphismGroups(Kernel(tC),Kernel(tD))=fail then
return fail;
fi;
if IsomorphismGroups(Kernel(sC),Kernel(sD))=fail then
return fail;
fi;
if IsomorphismGroups(Image(tC),Image(tD))=fail then
return fail;
fi;
if IsomorphismGroups(Image(sC),Image(sD))=fail then
return fail;
fi;
######################################################################
##
Map:=function()
xC:=[sC,tC];
xD:=ActToPair([sD,tD],f);
if xC=xD then
return InverseGeneralMapping(f);
fi;
############## Image of first component #####################
A:=AutomorphismGroup(G);
attr:=s->Image(s[1]);
actAttr:=ActToSubgroup;
processOrbit();
############## Kernel of first component #####################
attr:=s->Kernel(s[1]);
actAttr:=ActToSubgroup;
processOrbit();
if xC=xD then
return InverseGeneralMapping(f);
fi;
############## Image of second component #####################
attr:=s->Image(s[2]);
actAttr:=ActToSubgroup;
processOrbit();
if xC=xD then
return InverseGeneralMapping(f);
fi;
############## Kernel of second component #####################
attr:=s->Kernel(s[1]);
actAttr:=ActToSubgroup;
processOrbit();
if xC=xD then
return InverseGeneralMapping(f);
fi;
############## First component #####################
attr:=s->s[1];
actAttr:=ActToMap;
processOrbit();
if xC=xD then
return InverseGeneralMapping(f);
fi;
############## Second component #####################
attr:=s->s[2];
actAttr:=ActToMap;
processOrbit();
if xC=xD then
return InverseGeneralMapping(f);
fi;
return fail;
end;
##
########################################################################
map:=Map();
if map=fail then
return fail;
fi;
return Objectify(HapCatOneGroupMorphism,
rec(
source:= C,
target:= D,
mapping:= map
));
end);
##
##################### IsomorphismCatOneGroups ###############################