|
#############################################################################
##
#W pcppcgs.gi Polycyc Bettina Eick
#W Werner Nickel
##
#############################################################################
##
## At the moment the pcgs of a pcp group is called pcp. This is to keep
## it separated from the GAP library.
##
#############################################################################
##
#F UpdateCounter( ind, gens, c ) . . . . . . . . . . . . small help function
##
BindGlobal( "UpdateCounter", function( ind, gens, c )
local i, g;
# first reset c by ind
i := c - 1;
while i > 0 and not IsBool(ind[i]) and LeadingExponent(ind[i]) = 1 do
i := i - 1;
od;
if IsSortedList(gens) and not IsEmpty(gens) and
Depth(gens[Length(gens)]) < i then
return i + 1;
fi;
# now try to add elements from gens
repeat
g := First( gens, x -> Depth(x) = i and LeadingExponent(x) = 1 );
if not IsBool( g ) then
ind[i] := g;
i := i - 1;
fi;
until IsBool( g );
# return value for counter
return i + 1;
end );
#############################################################################
##
#F TailLimit
##
BindGlobal( "TailLimit", function( ind, c )
local k, i;
k := List(ind, x -> not IsBool(x) and LeadingExponent(x)=1);
i := c-1; while i > 0 and k[i]=true do i := i-1; od; i := i+1;
return i;
end );
#############################################################################
##
#F ReduceExpo
##
BindGlobal( "ReduceExpo", function( ind, gen, rel )
local i, j, a, b, q, f, k;
for i in [1..Length(ind)] do
if not IsBool(ind[i]) and rel[i]=0 then
b := LeadingExponent(ind[i]);
for j in [1..i-1] do
if not IsBool( ind[j] ) then
a := Exponents(ind[j])[i];
q := QuoInt(a,b);
if q <> 0 then
ind[j] := ind[j]*ind[i]^-q;
fi;
fi;
od;
for j in [1..Length(gen)] do
a := Exponents(gen[j])[i];
q := QuoInt(a,b);
if q <> 0 then
gen[j] := gen[j]*ind[i]^-q;
fi;
od;
fi;
od;
end );
#############################################################################
##
#F CheckIgs
##
BindGlobal( "CheckIgs", function( igs, gen )
local i, g, e, j;
for i in [1..Length(igs)] do
g := igs[i]^RelativeOrderPcp(igs[i]);
e := ExponentsByIgs(igs, g);
if e = fail then return i; fi;
for j in [1..i-1] do
g := Comm(igs[i], igs[j]);
e := ExponentsByIgs(igs,g);
if e = fail then return [i,j]; fi;
od;
od;
for i in [1..Length(gen)] do
e := ExponentsByIgs(igs, gen[i]);
if e = fail then return [i]; fi;
od;
return true;
end );
#############################################################################
##
#F ValFuns
##
IGSValFun1 := function(g) return 0; end;
IGSValFun2 := function(g) return AbsInt(LeadingExponent(g)); end;
IGSValFun3 := function(g)
return [AbsInt(LeadingExponent(g)),Length(Exponents(g))-Depth(g)];
end;
IGSValFun4 := function(g)
return [Length(Exponents(g))-Depth(g), AbsInt(LeadingExponent(g))];
end;
IGSValFun := IGSValFun4;
#############################################################################
##
#F AddToIgs( <igs>, <gens> )
##
InstallGlobalFunction(AddToIgs, function(igs, gens)
local coll, rels, n, c, ind, g, d, todo, val, j, f, h, e, a, k, b, u, t, r;
if Length(gens) = 0 then return igs; fi;
# get information
coll := Collector(gens[1]);
rels := RelativeOrders(coll);
n := NumberOfGenerators(coll);
c := n+1;
# set up
ind := ListWithIdenticalEntries(n, false);
for g in igs do ind[Depth(g)] := g; od;
# do a reduction step
c := TailLimit(ind, c);
todo := Set(Filtered(gens, x -> Depth(x) < c));
val := List(todo, x -> IGSValFun(x));
# loop over to-do list until it is empty
while Length(todo) > 0 and c > 1 do
j := Position(val, Minimum(val));
g := Remove(todo, j);
d := Depth(g);
f := [];
# shift g into ind
while d < c do
h := ind[d];
r := FactorOrder(g);
a := LeadingExponent(g);
# shift in
if IsBool(h) then
ind[d] := NormedPcpElement(g);
Add(f,d);
h := ind[d];
elif not IsPrime(r) then
b := LeadingExponent(h);
e := Gcdex(a, b);
if e.coeff1 <> 0 then
ind[d] := NormedPcpElement((g^e.coeff1)*(h^e.coeff2));
Add(f,d);
fi;
fi;
# divide off
if g = h then
g := g^0;
else
b := LeadingExponent(h);
e := Gcdex(a,b);
g := g^e.coeff3 * h^e.coeff4;
fi;
d := Depth(g);
od;
# adjust
c := TailLimit(ind, c);
ReduceExpo(ind, todo, rels);
# add powers and commutators
for d in f do
g := ind[d];
if rels[d] > 0 then
k := g ^ RelativeOrderPcp(g);
if Depth(k) < c then Add(todo, k); fi;
fi;
for j in [1..n] do
if not IsBool(ind[j]) then
k := Comm(g, ind[j]);
if Depth(k) < c then Add(todo, k); fi;
if rels[j] = 0 then
k := Comm(g, ind[j]^-1);
if Depth(k) < c then Add(todo, k); fi;
fi;
fi;
od;
od;
# reduce
todo := Filtered(todo, x -> Depth(x)<c);
val := List(todo, x -> IGSValFun(x));
Info(InfoPcpGrp, 3, Length(val)," versus ", ind);
od;
# return resulting list
ind := Filtered(ind, x -> not IsBool(x));
if CHECK_IGS@ then
Info(InfoPcpGrp, 1, "checking igs ");
t := CheckIgs(ind, gens);
if t <> true then Error("igs is incorrect at ",t); fi;
fi;
return ind;
end);
BindGlobal( "AddToIgs_Old", function(igs, gens)
local coll, rels, todo, n, ind, g, d, h, k, a, b, e, f, c, i, l;
if Length(gens) = 0 then return igs; fi;
# get information
coll := Collector(gens[1]);
rels := RelativeOrders(coll);
n := NumberOfGenerators(coll);
# create new list from igs
ind := ListWithIdenticalEntries(n, false);
for g in igs do ind[Depth(g)] := g; od;
# set counter and add tail as far as possible
c := UpdateCounter(ind, gens, n+1);
# create a to-do list and a pointer
todo := Set(Filtered(gens, x -> Depth(x) < c));
# loop over to-do list until it is empty
while Length(todo) > 0 and c > 1 do
g := Remove(todo);
d := Depth(g);
f := [];
# shift g into ind
while d < c do
h := ind[d];
if IsBool(h) then
ind[d] := g;
g := g^0;
Add(f, d);
else
# reduce g with h
a := LeadingExponent(g);
b := LeadingExponent(h);
e := Gcdex(a, b);
# adjust ind[d] by gcd
ind[d] := (g^e.coeff1) * (h^e.coeff2);
if e.coeff1 <> 0 then
Add(f, d);
fi;
# adjust g
g := (g^e.coeff3) * (h^e.coeff4);
fi;
d := Depth(g);
c := UpdateCounter(ind, todo, c);
od;
# add powers and commutators
for d in f do
g := ind[d];
if d <= Length(rels) and rels[d] > 0 and d < c then
k := g ^ RelativeOrderPcp(g);
if Depth(k) < c then Add(todo, k); fi;
fi;
for l in [1..n] do
if not IsBool(ind[l]) and (d < c or l < c) then
k := Comm(g, ind[l]);
if Depth(k) < c then Add(todo, k); fi;
k := Comm(g, ind[l]^-1);
if Depth(k) < c then Add(todo, k); fi;
fi;
od;
od;
# try sorting
Sort(todo);
od;
# return resulting list
return Filtered(ind, x -> not IsBool(x));
end );
#############################################################################
##
#F Igs( <gens> )
##
InstallOtherMethod( Igs, [IsList],
function( gens ) return AddToIgs( [], gens ); end );
#############################################################################
##
#F Ngs( <igs> ) . . . . . . . . . . . . . . . compute normed version of igs
##
InstallOtherMethod( Ngs, [IsList],
function( igs ) return List( igs, x -> NormedPcpElement( x ) ); end );
#############################################################################
##
#F Cgs( <igs> ) . . . . . .. . . . . . . . . compute canonical version of igs
##
InstallOtherMethod( Cgs, [IsList],
function( igs )
local ind, can, i, e, j, l, d, r, s;
# first norm leading coefficients
can := List( igs, x -> NormedPcpElement( x ) );
# reduce entries in matrix
for i in [1..Length(can)] do
e := LeadingExponent( can[i] );
d := Depth( can[i] );
for j in [1..i-1] do
l := Exponents( can[j] )[d];
if l > 0 then
r := QuoInt( l, e );
can[j] := can[j] * can[i]^-r;
elif l < 0 then
r := QuoInt( -l, e );
s := RemInt( -l, e );
if s = 0 then
can[j] := can[j] * can[i]^r;
else
can[j] := can[j] * can[i]^(r+1);
fi;
fi;
od;
od;
# set flag `normed' and return
for i in [1..Length(can)] do can[i]!.normed := true; od;
return can;
end );
#############################################################################
##
#F AddIgsToIgs( pcs1, pcs2 );
##
## Combines an igs <pcs2> of a normal subgroup with an igs <pcs1> of a
## factor. Typically, <pcs1> is induced wrt to a pcp and <pcs2> is the
## denominator of this pcp.
##
BindGlobal( "AddIgsToIgs", function( pcs1, pcs2 )
local coll, rels, n, ind, todo, g, c, h, eg, eh, e, d;
if Length( pcs1 ) = 0 then
return AsList( pcs2 );
elif Length( pcs2 ) = 0 then
return AsList( pcs1 );
elif Depth( pcs1[Length(pcs1)] ) < Depth( pcs2[1] ) then
return Concatenation( AsList( pcs1 ), AsList( pcs2 ) );
elif Depth( pcs2[Length(pcs2)] ) < Depth( pcs1[1] ) then
return Concatenation( AsList( pcs2 ), AsList( pcs1 ) );
fi;
# merge the two pcs'
coll := Collector( pcs1[1] );
rels := RelativeOrders( coll );
n := NumberOfGenerators( coll );
ind := List( [1..n], x -> false );
todo := [];
for g in pcs2 do ind[Depth(g)] := g; od;
for g in pcs1 do
if IsBool( ind[Depth(g)] ) then
ind[Depth(g)] := g;
else
Add( todo, g );
fi;
od;
# set counter
c := UpdateCounter( ind, todo, n+1 );
# create a to-do list and a pointer
todo := Filtered( todo, x -> Depth( x ) < c );
# loop over to-do list until it is empty
while Length( todo ) > 0 and c > 1 do
g := Remove(todo);
d := Depth( g );
# shift g into ind
while d < c do
h := ind[d];
if not IsBool( h ) then
# reduce g with h
eg := LeadingExponent( g );
eh := LeadingExponent( h );
e := Gcdex( eg, eh );
# adjust g and ind[d] by gcd
ind[d] := (g^e.coeff1) * (h^e.coeff2);
g := (g^e.coeff3) * (h^e.coeff4);
else
ind[d] := g;
g := g^0;
fi;
c := UpdateCounter( ind, todo, c );
d := Depth( g );
od;
od;
return Filtered( ind, x -> not IsBool( x ) );
end );
#############################################################################
##
#F ModuloInfo( igsH, igsN )
##
## igsH and igsN are igs'ses for H and N. We assume N <= H and N normal
## in H. The function computes information for the factor H/N.
##
BindGlobal( "ModuloInfo", function( igsH, igsN )
local depN, gens, rels, h, r, j, l, e;
depN := List( igsN, Depth );
gens := [];
rels := [];
# get modulo generators and their relative orders
for h in igsH do
r := RelativeOrderPcp( h );
j := PositionSet( depN, Depth(h) );
if IsBool( j ) then
Add( rels, r );
Add( gens, h );
elif r > 0 then
if not IsPrime( r ) then
l := RelativeOrderPcp( igsN[j] );
if l <> r then
Add( rels, r / l );
Add( gens, h );
fi;
fi;
else
e := AbsInt( LeadingExponent( igsN[j] ) / LeadingExponent( h ) );
if e > 1 then
Add( rels, e );
Add( gens, h );
fi;
fi;
od;
return rec( gens := gens, rels := rels );
end );
#############################################################################
##
#F CyclicDecomposition( pcp )
##
BindGlobal( "CyclicDecomposition", function( pcp )
local rels, n, mat, i, row, new, cyc, ord, chg, inv, g, tmp, imgs, prei;
# catch a trivial case
if Length( pcp ) = 0 then
return rec( gens := [], rels := [], chg := [], inv := [] );
fi;
# set up
rels := RelativeOrdersOfPcp( pcp );
n := Length( pcp );
# create relator matrix for power relators - this is in upper
# triangular form
mat := [];
for i in [1..n] do
if rels[i] > 0 then
row := ExponentsByPcp( pcp, pcp[i]^rels[i] );
row[i] := row[i] - rels[i];
Add( mat, row );
else
Add( mat, List( [1..n], x -> 0 ) );
fi;
od;
# solve matrix
# new := SmithNormalFormSQ( mat );
new := NormalFormIntMat( mat, 9 );
# get new generators, relators and the basechange
cyc := [];
ord := [];
chg := [];
inv := [];
imgs := TransposedMat( new.coltrans );
prei := Inverse( new.coltrans );
for i in [1..n] do
if new.normal[i][i] <> 1 then
g := MappedVector( prei[i], pcp );
Add( cyc, g );
Add( ord, new.normal[i][i] );
Add( chg, prei[i] );
if new.normal[i][i] > 0 then
Add( inv, List( imgs[i], x -> x mod new.normal[i][i] ) );
else
Add( inv, imgs[i] );
fi;
fi;
od;
return rec( gens := cyc,
rels := ord,
chg := chg,
inv := TransposedMat( inv ) );
end );
#############################################################################
##
#F AddTailInfo( pcp ) . . . . . . .
##
## The info in pcp!.tail is used to compute exponent vectors.
## 1.) pcp!.tail is a list, then exponents are just looked up.
## 2.) pcp!.tail is an integer, then the computation of exponents
## stops at pcp!.tail-1;
##
BindGlobal( "AddTailInfo", function( pcp )
local gens, sub, n, deps, depg, i, d, mult;
gens := pcp!.gens;
sub := pcp!.denom;
# if there are no gens, then it does not matter
if Length( gens ) = 0 then return; fi;
n := NumberOfGenerators( Collector( gens[1] ) );
# get depths
deps := List( sub, Depth );
depg := List( gens, Depth );
# if not IsSortedList( deps ) then Error("add tail info"); fi;
# set tail to an integer
pcp!.tail := Maximum( depg ) + 1;
# FIXME: the remainder of this function does not what it is supposed to
# do, so we skip it for now
return;
if not IsSortedList( deps ) then return; fi;
# now figure out whether we can do better
for i in [1..Length(sub)] do
if deps[i] < pcp!.tail - 1 then
d := IsPowerOfGenerator( sub[i], pcp!.tail );
if IsBool( d ) then return; fi;
fi;
od;
# add multiplication list
mult := [];
for i in [1..Length(gens)] do
if depg[i] < pcp!.tail then
d := IsPowerOfGenerator( gens[i], pcp!.tail );
if IsBool( d ) then return; fi;
Add( mult, d );
fi;
od;
# if we arrive here, then we may read off exponents
pcp!.tail := depg;
if ForAny( mult, x -> x <> 1 ) then pcp!.mult := mult; fi;
end );
#############################################################################
##
#F Creation function for pcp's.
##
## Pcp( U ) pcp for U
## Pcp( U, N ) pcp for U mod N
## Pcp( U, "snf" ) pcp for abelian group U in SNF
## Pcp( U, N, "snf" ) pcp for abelian factor U mod N in SNF
##
InstallGlobalFunction( Pcp, function( arg )
local U, gens, rels, denom, numer, info, pcp;
# catch arguments U and N
U := arg[1];
if not IsPcpGroup(U) then
Error("<U> must be a pcp group");
fi;
if Length( arg ) = 1 or IsString( arg[2] ) then
denom := [];
elif Length( arg ) > 1 and IsGroup( arg[2] ) then
denom := arg[2];
fi;
# do we want to norm the pcs or make it canonical?
if USE_CANONICAL_PCS@ then
numer := Cgs( U );
denom := Cgs( denom );
elif USE_NORMED_PCS@ then
numer := Ngs( U );
denom := Ngs( denom );
else
numer := Igs( U );
denom := Igs( denom );
fi;
# set up modulo info
if Length( denom ) > 0 then
info := ModuloInfo( numer, denom );
gens := info.gens;
rels := info.rels;
else
gens := numer;
rels := List( gens, RelativeOrderPcp );
fi;
# create pcp record and objectify
pcp := rec( gens := gens,
rels := rels,
denom := denom,
numer := numer,
one := One( U ),
group := U );
pcp := Objectify( PcpType, pcp );
# add info on tails
AddTailInfo( pcp );
# add info on snf if desired
if arg[Length(arg)] = "snf" then
pcp!.cyc := CyclicDecomposition( pcp );
fi;
# return
return pcp;
end );
#############################################################################
##
#F Basic attributes and properties - for IsPcpRep
##
InstallGlobalFunction( RelativeOrdersOfPcp, function( pcp )
if IsBound( pcp!.cyc ) then
return pcp!.cyc.rels;
else
return pcp!.rels;
fi;
end );
InstallGlobalFunction( GeneratorsOfPcp, function( pcp )
if IsBound( pcp!.cyc ) then
return pcp!.cyc.gens;
else
return pcp!.gens;
fi;
end );
InstallGlobalFunction( DenominatorOfPcp, pcp -> pcp!.denom );
InstallGlobalFunction( NumeratorOfPcp, pcp -> pcp!.numer );
InstallGlobalFunction( OneOfPcp, pcp -> pcp!.one );
InstallGlobalFunction( GroupOfPcp, pcp -> pcp!.group );
InstallGlobalFunction( IsSNFPcp, pcp -> IsBound(pcp!.cyc) );
InstallGlobalFunction( IsTailPcp, pcp -> IsList(pcp!.tail) );
#############################################################################
##
#F Higher-level attributes and properties - to make pcp's look like lists
##
#############################################################################
##
#M Length( <pcp> )
##
InstallOtherMethod( Length, [ IsPcp ],
pcp -> Length( GeneratorsOfPcp( pcp ) ) );
#############################################################################
##
#M AsList( <pcp> )
##
InstallOtherMethod( AsList, [ IsPcp ],
pcp -> GeneratorsOfPcp( pcp ) );
#############################################################################
##
#M Position( <pcp>, <elm>, <from> )
##
InstallOtherMethod( Position,
[ IsPcp, IsPcpElement, IsInt ],
function( pcp, elm, from )
return Position( AsList( pcp ), elm, from );
end );
#############################################################################
##
#M ListOp( pcp, function )
##
InstallOtherMethod( ListOp,
[ IsPcp, IsObject ],
function( pcp, f )
return List( AsList(pcp), f );
end );
#############################################################################
##
#M <pcp> [ <pos> ]
##
InstallOtherMethod( \[\],
[ IsPcp, IsPosInt ],
function( pcp, pos )
return GeneratorsOfPcp(pcp)[pos];
end );
#############################################################################
##
#M <pcp>{[ <pos> ]}
##
InstallOtherMethod( ELMS_LIST, [ IsPcp, IsDenseList ],
function( pcp, ran )
return GeneratorsOfPcp( pcp ){ran};
end );
#############################################################################
##
#M Print pcp
##
InstallMethod( PrintObj, "for pcp", [IsPcp],
function( pcp )
Print( "Pcp ", GeneratorsOfPcp( pcp ), " with orders ",
RelativeOrdersOfPcp(pcp));
end );
InstallMethod( ViewObj, [ IsPcp ], SUM_FLAGS, PrintObj );
#############################################################################
##
#F small helper
##
BindGlobal( "WordByExps@", function( exp )
local w, i;
w := [];
for i in [1..Length(exp)] do
if exp[i] <> 0 then
Add( w, i );
Add( w, exp[i] );
fi;
od;
return w;
end );
#############################################################################
##
#M a small helper
##
BindGlobal( "PrintWord", function(gen,exp)
local w, i, g;
w := WordByExps@(exp);
if Length(w) = 0 then
Print("id ");
else
for i in [1,3..Length(w)-1] do
g := Concatenation(gen,String(w[i]));
if w[i+1] = 1 then
Print(g);
else
Print(g,"^",w[i+1]);
fi;
if i < Length(w)-1 then
Print(" * ");
fi;
od;
fi;
Print("\n");
end );
#############################################################################
##
#M Print pcp presentation
##
BindGlobal( "PrintPresentationByPcp", function( pcp, flag )
local gens, rels, i, r, g, j, h, c;
gens := GeneratorsOfPcp( pcp );
rels := RelativeOrdersOfPcp( pcp );
# print relations
for i in [1..Length(gens)] do
if rels[i] > 0 then
r := rels[i];
g := gens[i];
Print("g",i,"^",r," = ");
PrintWord("g",ExponentsByPcp(pcp, g^r));
fi;
od;
for i in [1..Length(gens)] do
for j in [1..i-1] do
g := gens[i];
h := gens[j];
c := gens[i]^gens[j];
if c <> g or flag = "all" then
Print("g",i," ^ g",j," = ");
PrintWord("g",ExponentsByPcp(pcp, c));
fi;
if rels[j] = 0 or flag = "all" then
c := gens[i]^(gens[j]^-1);
if c <> g or flag = "all" then
Print("g",i," ^ g",j,"^-1 = ");
PrintWord("g",ExponentsByPcp(pcp, c));
fi;
fi;
od;
od;
end );
#############################################################################
##
#M Print pcp presentation
##
BindGlobal( "PrintPcpPresentation", function( arg )
local G, flag;
G := arg[1];
if Length(arg) = 2 then
flag := arg[2];
else
flag := false;
fi;
if IsGroup(G) then
PrintPresentationByPcp( Pcp(G), flag );
else
PrintPresentationByPcp( G, flag );
fi;
end );
#############################################################################
##
#M GapInputPcpGroup( file, pcp )
##
BindGlobal( "GapInputPcpGroup", function( file, pcp )
local gens, rels, i, j, obj;
gens := GeneratorsOfPcp( pcp );
rels := RelativeOrdersOfPcp( pcp );
PrintTo(file, "coll := FromTheLeftCollector( ", Length(gens)," );\n");
for i in [1..Length(rels)] do
if rels[i] > 0 then
obj := WordByExps@(ExponentsByPcp( pcp, gens[i]^rels[i] ));
AppendTo(file, "SetRelativeOrder( coll, ",i,", ",rels[i]," );\n");
AppendTo(file, "SetPower( coll, ",i,", ",obj," );\n");
fi;
od;
for i in [1..Length(rels)] do
for j in [1..i-1] do
obj := WordByExps@(ExponentsByPcp( pcp, gens[i]^gens[j] ));
if obj <> [ i, 1 ] then
AppendTo(file,
"SetConjugate( coll, ",i,", ",j,", ",obj," );\n");
fi;
obj := WordByExps@(ExponentsByPcp( pcp, gens[i]^(gens[j]^-1) ));
if obj <> [ i, 1 ] then
AppendTo(file,
"SetConjugate( coll, ",i,", ",-j,", ",obj," );\n");
fi;
od;
od;
AppendTo(file, "UpdatePolycyclicCollector( coll );\n" );
AppendTo(file, "G := PcpGroupByCollectorNC( coll ); \n");
if HasIsNilpotentGroup( GroupOfPcp(pcp) ) and
IsNilpotentGroup( GroupOfPcp(pcp) ) then
AppendTo(file, "SetIsNilpotentGroup( G, true );\n" );
fi;
end );
#############################################################################
##
#M PcpGroupByPcp( pcp ) . . . . . . . . . . . . . . . . . create a new group
##
BindGlobal( "PcpGroupByPcp", function( pcp )
local g, r, n, coll, i, j, h, e, w, G;
# write down a presentation
g := GeneratorsOfPcp( pcp );
r := RelativeOrdersOfPcp( pcp );
n := Length( g );
# create a collector
coll := FromTheLeftCollector( n );
for i in [1..n] do
if r[i] > 0 then
SetRelativeOrder( coll, i, r[i] );
h := g[i] ^ r[i];
e := ExponentsByPcp( pcp, h );
w := ObjByExponents( coll, e );
if Length( w ) > 0 then SetPower( coll, i, w ); fi;
fi;
for j in [1..i-1] do
h := g[i]^g[j];
e := ExponentsByPcp( pcp, h );
w := ObjByExponents( coll, e );
if Length( w ) > 0 then SetConjugate( coll, i, j, w ); fi;
h := g[i]^(g[j]^-1);
e := ExponentsByPcp( pcp, h );
w := ObjByExponents( coll, e );
if Length( w ) > 0 then SetConjugate( coll, i, -j, w ); fi;
od;
od;
UpdatePolycyclicCollector( coll );
G := PcpGroupByCollectorNC( coll );
return G;
end );
BindGlobal( "DisplayPcpGroup", function( G )
local collector, gens, rods, n, g, h, conj;
collector := Collector( G );
gens := Pcp( G );
rods := RelativeOrdersOfPcp( gens );
n := Length( gens );
Print( "<" );
for g in [1..n] do Print( " ", gens[g] ); od;
Print( " | \n\n" );
for g in [1..n] do
if rods[g] <> 0 then
## print the power relation for g.
Print( " ", gens[g], "^", rods[g], " = ",
gens[g]^rods[g], "\n" );
fi;
od;
if rods <> 0 * rods then Print( "\n" ); fi;
for h in [1..n] do
for g in [1..h-1] do
conj := gens[h]^gens[g];
if conj <> gens[h] then
## print the conjuagte relation for h^g.
Print( " ", gens[h], "^", gens[g], " = ",
gens[h]^gens[g], "\n" );
fi;
od;
od;
Print( ">\n" );
end );
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