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############################################################################
##
#W initqs.gi LPRES René Hartung
##
############################################################################
##
#M InitQuotientSystem ( <LpGroup> )
##
## computes a weighted nilpotent quotient system for the abelian quotient
## of <LpGroup>.
##
InstallMethod( InitQuotientSystem,
"for an L-presented group",
true,
[ IsLpGroup ], 0,
function(G)
local ftl, # FromTheLeftCollector for G/G'
A, # power relations from Hermite normal form
Q, # new quotient system
n, # number of generators of L
ev,evn, # exponent vectors for spinning algorithm
rel, # loop variable for (iterated) relators
map, # loop variable for endomorphisms
i,j,k, # loop variables
HNF, # Hermite normal form of the relations
stack, # stack for the spinning algorithm
endos, # endomorphisms as matrices
obj, mat, # loop variables to determine the matrices
Gens, # position of new gens in the HNF
Imgs; # loop variable to build the endomorphism
# number of new generators
n:=Length(GeneratorsOfGroup(G));
# determine the iterated relators
stack:=[];
HNF:=rec(mat:=[],Heads:=[]);
for rel in IteratedRelatorsOfLpGroup(G) do
ev:=ListWithIdenticalEntries(n,0);
obj:=ExtRepOfObj(rel);
for i in [1,3..Length(obj)-1] do
ev[obj[i]]:=ev[obj[i]]+obj[i+1];
od;
Add(stack,ShallowCopy(ev));
LPRES_AddRow(HNF,ev);
od;
# determine the endomorphisms
endos:=[];
for map in EndomorphismsOfLpGroup(G) do
mat:=NullMat(n,n);
# UnderlyingElement: map is an endomorphism of the free group;
# x an object of the L-presented group
obj:=List(GeneratorsOfGroup(G),x->ExtRepOfObj(UnderlyingElement(x)^map));
for j in [1..n] do
for k in [1,3..Length(obj[j])-1] do
mat[j][obj[j][k]]:=mat[j][obj[j][k]]+obj[j][k+1];
od;
od;
Add(endos,mat);
od;
# spinning algorithm
while not Length(stack)=0 do
ev:=stack[1];
Remove( stack, 1);
if not IsZero(ev) then
for i in [1..Length(endos)] do
evn:=ev*endos[i];
if LPRES_AddRow(HNF,evn) then
Add(stack,evn);
fi;
od;
fi;
od;
# add the (fixed) relators
for rel in FixedRelatorsOfLpGroup(G) do
ev:=ListWithIdenticalEntries(n,0);
obj:=ExtRepOfObj(rel);
for i in [1,3..Length(obj)-1] do
ev[obj[i]]:=ev[obj[i]]+obj[i+1];
od;
LPRES_AddRow(HNF,ev);
od;
# translate the Hermite normal form to power relations
A:=LPRES_PowerRelationsOfHNF(HNF);
# generators with power relation
Gens:=HNF.Heads{Filtered([1..Length(HNF.Heads)],
x->HNF.mat[x][HNF.Heads[x]]<>1)};
# infinite generators;
Append(Gens,Filtered([1..n],x->not x in HNF.Heads));
Sort(Gens);
# if the group is perfect
if Length(Gens)=0 then
Q := rec( Lpres := G,
Pccol := FromTheLeftCollector( 0 ),
Imgs := ListWithIdenticalEntries( n, [] ),
Weights := [],
Definitions := [] );
UpdatePolycyclicCollector( Q.Pccol );
Imgs := List( Q.Imgs, x -> PcpElementByGenExpList( Q.Pccol, x ) );
Q.Epimorphism := GroupHomomorphismByImagesNC( Q.Lpres,
PcpGroupByCollectorNC( Q.Pccol),
GeneratorsOfGroup( Q.Lpres), Imgs );
return( Q );
fi;
# build quotient system
Q:=rec();
Q.Lpres:=G;
# the new collector
Q.Pccol:=FromTheLeftCollector(Length(Gens));
for i in [1..Length(Gens)] do
k:=Position(HNF.Heads,Gens[i]);
if k<>fail then
SetRelativeOrder(Q.Pccol,i, A[k][Gens[i]]);
ev:=ShallowCopy(A[k]{Gens});
ev[i]:=0;
if not IsZero(ev) then
SetPower(Q.Pccol,i,ObjByExponents(Q.Pccol,-ev));
fi;
fi;
od;
UpdatePolycyclicCollector(Q.Pccol);
# the natural homomorphism onto the new presentation
Q.Imgs:=[];
for i in [1..n] do
if i in Gens then
k:=i-Length(Filtered([1..Length(HNF.Heads)],x->HNF.mat[x][HNF.Heads[x]]=1
and HNF.Heads[x]<i));
Add(Q.Imgs,k);
else
Add(Q.Imgs,ObjByExponents(Q.Pccol,-A[Position(HNF.Heads,i)]{Gens}));
fi;
od;
# the epimorphism into the new presentation
Imgs:=[];
for i in [1..Length(Q.Imgs)] do
if IsInt(Q.Imgs[i]) then
Add(Imgs,PcpElementByGenExpList(Q.Pccol,[Q.Imgs[i],1]));
else
Add(Imgs,PcpElementByGenExpList(Q.Pccol,Q.Imgs[i]));
fi;
od;
Q.Epimorphism:=GroupHomomorphismByImagesNC(Q.Lpres,
PcpGroupByCollectorNC(Q.Pccol),
GeneratorsOfGroup(Q.Lpres),Imgs);
Q.Weights:=List([1..Length(Gens)],x->1);
Q.Definitions:=Filtered([1..Length(Q.Imgs)],x->not IsList(Q.Imgs[x]));
return(Q);
end);
############################################################################
##
#M AbelianInvariants ( <LpGroup> )
##
## computes the abelian invariants of <LpGroup>.
##
InstallMethod( AbelianInvariants,
"for an L-presented group",
[ IsLpGroup ],0,
G -> AbelianInvariants( NilpotentQuotient(G,1) ) );
[ Dauer der Verarbeitung: 0.32 Sekunden
(vorverarbeitet)
]
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