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#############################################################################
##
#W pcpgrps.gi Polycyc Bettina Eick
##
#############################################################################
##
#F Create a pcp group by collector
##
## The trivial group is missing.
##
InstallGlobalFunction( PcpGroupByCollectorNC, function( coll )
local n, l, e, f, G, rels;
n := coll![ PC_NUMBER_OF_GENERATORS ];
if n > 0 then
l := IdentityMat( n );
fi;
e := List( [1..n], x -> PcpElementByExponentsNC( coll, l[x] ) );
f := PcpElementByGenExpListNC( coll, [] );
G := GroupWithGenerators( e, f );
SetCgs( G, e );
SetIsWholeFamily( G, true );
if 0 in RelativeOrders( coll ) then
SetIsFinite( G, false );
else
SetIsFinite( G, true );
SetSize( G, Product( RelativeOrders( coll ) ) );
fi;
return G;
end );
InstallGlobalFunction( PcpGroupByCollector, function( coll )
UpdatePolycyclicCollector( coll );
if not IsConfluent( coll ) then
return fail;
else
return PcpGroupByCollectorNC( coll );
fi;
end );
#############################################################################
##
#F Print( <G> )
##
InstallMethod( PrintObj, "for a pcp group", [ IsPcpGroup ],
function( G )
Print("Pcp-group with orders ", List( Igs(G), RelativeOrderPcp ) );
end );
InstallMethod( ViewObj, "for a pcp group", [ IsPcpGroup ], SUM_FLAGS,
function( G )
Print("Pcp-group with orders ", List( Igs(G), RelativeOrderPcp ) );
end );
#############################################################################
##
#M Igs( <pcpgrp> )
#M Ngs( <pcpgrp> )
#M Cgs( <pcpgrp> )
##
InstallMethod( Igs, [ IsPcpGroup ],
function( G )
if HasCgs( G ) then return Cgs(G); fi;
if HasNgs( G ) then return Ngs(G); fi;
return Igs( GeneratorsOfGroup(G) );
end );
InstallMethod( Ngs, [ IsPcpGroup ],
function( G )
if HasCgs( G ) then return Cgs(G); fi;
return Ngs( Igs( GeneratorsOfGroup(G) ) );
end );
InstallMethod( Cgs, [ IsPcpGroup ],
function( G )
return Cgs( Igs( GeneratorsOfGroup(G) ) );
end );
#############################################################################
##
#M Membershiptest for pcp groups
##
InstallMethod( \in, "for a pcp element and a pcp group",
IsElmsColls, [IsPcpElement, IsPcpGroup],
function( g, G )
return ReducedByIgs( Igs(G), g ) = One(G);
end );
#############################################################################
##
#M Random( G )
##
InstallMethodWithRandomSource( Random, "for a random source and a pcp group",
[ IsRandomSource, IsPcpGroup ],
function( rs, G )
local pcp, rel, g, i;
pcp := Pcp(G);
if Length( pcp ) = 0 then
return One( G );
fi;
rel := RelativeOrdersOfPcp( pcp );
g := [];
for i in [1..Length(rel)] do
if rel[i] = 0 then
g[i] := Random( rs, Integers );
else
g[i] := Random( rs, 0, rel[i]-1 );
fi;
od;
return MappedVector( g, pcp );
end );
#############################################################################
##
#M SubgroupByIgs( G, igs [, gens] )
##
## create a subgroup and set igs. If gens is given, compute pcs defined
## by <gens, pcs> and use this. Note: this function does not check if the
## generators are in G.
##
InstallGlobalFunction( SubgroupByIgs,
function( arg )
local U, pcs;
if Length( arg ) = 3 then
pcs := AddToIgs( arg[2], arg[3] );
else
pcs := arg[2];
fi;
U := SubgroupNC( Parent(arg[1]), pcs );
SetIgs( U, pcs );
return U;
end);
#############################################################################
##
#M SubgroupByIgsAndIgs( G, fac, nor )
##
## fac and nor are igs for a factor and a normal subgroup. This function
## computes an igs for the subgroups generated by fac and nor and sets it
## in the subgroup. This is a no-check function
##
BindGlobal( "SubgroupByIgsAndIgs", function( G, fac, nor )
return SubgroupByIgs( G, AddIgsToIgs( fac, nor ) );
end );
#############################################################################
##
#M IsSubset for pcp groups
##
InstallMethod( IsSubset, "for pcp groups",
IsIdenticalObj, [ IsPcpGroup, IsPcpGroup ], SUM_FLAGS,
function( H, U )
if Parent(U) = H then return true; fi;
return ForAll( GeneratorsOfGroup(U), x -> x in H );
end );
#############################################################################
##
#M Size( <pcpgrp> )
##
InstallMethod( Size, [ IsPcpGroup ],
function( G )
local pcs, rel;
pcs := Igs( G );
rel := List( pcs, RelativeOrderPcp );
if ForAny( rel, x -> x = 0 ) then
return infinity;
else
return Product( rel );
fi;
end );
#############################################################################
##
#M CanComputeIndex( <pcpgrp>, <pcpgrp> )
#M CanComputeSize( <pcpgrp> )
#M CanComputeSizeAnySubgroup( <pcpgrp> )
##
InstallMethod( CanComputeIndex, IsIdenticalObj, [IsPcpGroup, IsPcpGroup], ReturnTrue );
InstallTrueMethod( CanComputeSize, IsPcpGroup );
InstallTrueMethod( CanComputeSizeAnySubgroup, IsPcpGroup );
#############################################################################
##
#M IndexNC/Index( <pcpgrp>, <pcpgrp> )
##
InstallMethod( IndexNC, "for pcp groups",
IsIdenticalObj, [IsPcpGroup, IsPcpGroup],
function( H, U )
local pcp, rel;
pcp := Pcp( H, U );
rel := RelativeOrdersOfPcp( pcp );
if ForAny( rel, x -> x = 0 ) then
return infinity;
else
return Product( rel );
fi;
end );
InstallMethod( IndexOp, "for pcp groups",
IsIdenticalObj, [IsPcpGroup, IsPcpGroup],
function( H, U )
if not IsSubgroup( H, U ) then
Error("H must be contained in G");
fi;
return IndexNC( H, U );
end );
#############################################################################
##
#M <pcpgrp> = <pcpgrp>
##
InstallMethod( \=, "for pcp groups",
IsIdenticalObj, [IsPcpGroup, IsPcpGroup],
function( G, H )
return Cgs( G ) = Cgs( H );
end);
#############################################################################
##
#M ClosureGroup( <pcpgrp>, <pcpgrp> )
##
InstallMethod( ClosureGroup, "for pcp groups",
IsIdenticalObj, [IsPcpGroup, IsPcpGroup],
function( G, H )
local P;
P := PcpGroupByCollectorNC( Collector(G) );
return SubgroupByIgs( P, Igs(G), GeneratorsOfGroup(H) );
end );
#############################################################################
##
#F HirschLength( <G> )
##
InstallMethod( HirschLength, [ IsPcpGroup ],
function( G )
local pcs, rel;
pcs := Igs( G );
rel := List( pcs, RelativeOrderPcp );
return Length( Filtered( rel, x -> x = 0 ) );
end );
#############################################################################
##
#M CommutatorSubgroup( G, H )
##
InstallMethod( CommutatorSubgroup, "for pcp groups",
IsIdenticalObj, [ IsPcpGroup, IsPcpGroup],
function( G, H )
local pcsG, pcsH, coms, i, j, U, u;
pcsG := Igs(G);
pcsH := Igs(H);
# if G = H then we need fewer commutators
coms := [];
if pcsG = pcsH then
for i in [1..Length(pcsG)] do
for j in [1..i-1] do
Add( coms, Comm( pcsG[i], pcsH[j] ) );
od;
od;
coms := Igs( coms );
U := SubgroupByIgs( Parent(G), coms );
else
for u in pcsG do
coms := AddToIgs( coms, List( pcsH, x -> Comm( u, x ) ) );
od;
U := SubgroupByIgs( Parent(G), coms );
# In order to conjugate with fewer elements, compute <U,V>. If one is
# normal than we do not need the normal closure, see Glasby 1987.
if not (IsBound( G!.isNormal ) and G!.isNormal) and
not (IsBound( H!.isNormal ) and H!.isNormal) then
U := NormalClosure( ClosureGroup( G, H ), U );
fi;
fi;
return U;
end );
#############################################################################
##
#M DerivedSubgroup( G )
##
InstallMethod( DerivedSubgroup, "for a pcp group",
[ IsPcpGroup ], G -> CommutatorSubgroup(G, G) );
#############################################################################
##
#M PRump( G, p ). . . . . smallest normal subgroup N of G with G/N elementary
## abelian p-group.
##
InstallMethod( PRumpOp, "for a pcp group and a prime",
[IsPcpGroup, IsPosInt],
function( G, p )
local D, pcp, new;
D := DerivedSubgroup(G);
pcp := Pcp( G, D );
new := List( pcp, x -> x^p );
return SubgroupByIgs( G, Igs(D), new );
end );
#############################################################################
##
#M IsNilpotentGroup( <pcpgrp> )
##
InstallMethod( IsNilpotentGroup, "for a pcp group with known lower central series",
[ IsPcpGroup and HasLowerCentralSeriesOfGroup ],
function( G )
local lcs;
lcs := LowerCentralSeriesOfGroup( G );
return IsTrivial( lcs[ Length(lcs) ] );
end );
InstallMethod( IsNilpotentGroup, "for a pcp group",
[IsPcpGroup],
function( G )
local l, U, V, pcp, n;
l := HirschLength(G);
U := ShallowCopy( G );
repeat
# take next term of lc series
U!.isNormal := true;
V := CommutatorSubgroup( G, U );
# if we arrive at the trivial group
if Size( V ) = 1 then return true; fi;
# get quotient U/V
pcp := Pcp( U, V );
# if U=V then the series has terminated at a non-trivial group
if Length( pcp ) = 0 then
return false;
fi;
# get the Hirsch length of U/V
n := Length( Filtered( RelativeOrdersOfPcp( pcp ), x -> x = 0));
# compare it with l
if n = 0 and l <> 0 then return false; fi;
l := l - n;
# iterate
U := ShallowCopy( V );
until false;
end );
#############################################################################
##
#M IsElementaryAbelian( <pcpgrp> )
##
InstallMethod( IsElementaryAbelian, "for a pcp group",
[ IsPcpGroup ],
function( G )
local rel, p;
if not IsFinite(G) or not IsAbelian(G) then return false; fi;
rel := List( Igs(G), RelativeOrderPcp );
if Length(Set(rel)) > 1 then return false; fi;
if ForAny( rel, x -> not IsPrime(x) ) then return false; fi;
p := rel[1];
return ForAll( RelativeOrdersOfPcp( Pcp( G, "snf" ) ), x -> x = p );
end );
#############################################################################
##
#F AbelianInvariants( <pcpgrp > )
##
InstallMethod( AbelianInvariants, "for a pcp group",
[ IsPcpGroup ],
function( G )
return AbelianInvariantsOfList( RelativeOrdersOfPcp( Pcp(G, DerivedSubgroup(G), "snf") ) );
end );
InstallMethod( AbelianInvariants, "for an abelian pcp group",
[IsPcpGroup and IsAbelian],
function( G )
return AbelianInvariantsOfList( RelativeOrdersOfPcp( Pcp(G, "snf") ) );
end );
#############################################################################
##
#M CanEasilyComputeWithIndependentGensAbelianGroup( <pcpgrp> )
##
if IsBound(CanEasilyComputeWithIndependentGensAbelianGroup) then
# CanEasilyComputeWithIndependentGensAbelianGroup was introduced in GAP 4.5.x
InstallTrueMethod(CanEasilyComputeWithIndependentGensAbelianGroup,
IsPcpGroup and IsAbelian);
fi;
BindGlobal( "ComputeIndependentGeneratorsOfAbelianPcpGroup", function ( G )
local pcp, id, mat, base, ord, i, g, o, cf, j;
# Get a pcp in Smith normal form
if not IsBound( G!.snfpcp ) then
pcp := Pcp(G, "snf");
G!.snfpcp := pcp;
else
pcp := G!.snfpcp;
fi;
if IsBound( G!.indgens ) and IsBound( G!.indgenmat ) then
return;
fi;
# Unfortunately, this is not *quite* what we need; in order to match
# the Abelian invariants, we now have to further refine the generator
# list to ensure only generators of prime power order are in the list.
id := IdentityMat( Length(pcp) );
mat := [];
base := [];
ord := [];
for i in [1..Length(pcp)] do
g := pcp[i];
o := Order(g);
if o = 1 then continue; fi;
if o = infinity then
Add(base, g);
Add(mat, id[i]);
Add(ord, 0);
continue;
fi;
cf:=Collected(Factors(o));
if Length(cf) > 1 then
for j in cf do
j := j[1]^j[2];
Add(base, g^(o/j));
Add(mat, id[i] * (j/o mod j));
Add(ord, j);
od;
else
Add(base, g);
Add(mat, id[i]);
Add(ord, o);
fi;
od;
SortParallel(ShallowCopy(ord),base);
SortParallel(ord,mat);
mat := TransposedMat( mat );
G!.indgens := base;
G!.indgenmat := mat;
end );
#############################################################################
##
#A IndependentGeneratorsOfAbelianGroup( <A> )
##
InstallMethod(IndependentGeneratorsOfAbelianGroup, "for an abelian pcp group",
[IsPcpGroup and IsAbelian],
function( G )
if not IsBound( G!.indgens ) then
ComputeIndependentGeneratorsOfAbelianPcpGroup( G );
fi;
return G!.indgens;
end );
#############################################################################
##
#O IndependentGeneratorExponents( <G>, <g> )
##
if IsBound( IndependentGeneratorExponents ) then
# IndependentGeneratorExponents was introduced in GAP 4.5.x
InstallMethod(IndependentGeneratorExponents, "for an abelian pcp group and an element",
IsCollsElms,
[IsPcpGroup and IsAbelian, IsPcpElement],
function( G, elm )
local exp, rels, i;
# Ensure everything has been set up
if not IsBound( G!.indgenmat ) then
ComputeIndependentGeneratorsOfAbelianPcpGroup( G );
fi;
# Convert elm into an exponent vector with respect to a snf pcp
exp := ExponentsByPcp( G!.snfpcp, elm );
rels := AbelianInvariants( G );
# Convert the exponent vector with respect to pcp into one
# with respect to our independent abelian generators.
exp := exp * G!.indgenmat;
for i in [1..Length(exp)] do
if rels[i] > 0 then exp[i] := exp[i] mod rels[i]; fi;
od;
return exp;
end);
fi;
#############################################################################
##
#F NormalClosure( K, U )
##
InstallMethod( NormalClosureOp, "for pcp groups",
IsIdenticalObj, [IsPcpGroup, IsPcpGroup],
function( K, U )
local tmpN, newN, done, id, gensK, pcsN, k, n, c, N;
# take initial pcs
pcsN := ShallowCopy( Cgs(U) );
if Length( pcsN ) = 0 then return U; fi;
# take generating sets
id := One( K );
gensK := GeneratorsOfGroup(K);
gensK := List( gensK, x -> ReducedByIgs( pcsN, x ) );
gensK := Filtered( gensK, x -> x <> id );
repeat
done := true;
tmpN := ShallowCopy( pcsN );
for k in gensK do
for n in tmpN do
c := ReducedByIgs( pcsN, Comm( k, n ) );
if c <> id then
newN := AddToIgs( pcsN, [c] );
if newN <> pcsN then
done := false;
pcsN := Cgs( newN );
fi;
fi;
od;
od;
#Print(Length(pcsN)," obtained \n");
until done;
# set up result
N := Group( pcsN );
SetIgs( N, pcsN );
return N;
end);
#############################################################################
##
#F ExponentsByRels . . . . . . . . . . . . . . .elements written as exponents
##
BindGlobal( "ExponentsByRels", function( rel )
local exp, idm, i, t, j, e, f;
exp := [List( rel, x -> 0 )];
idm := IdentityMat( Length( rel ) );
for i in Reversed( [1..Length(rel)] ) do
t := [];
for j in [1..rel[i]] do
for e in exp do
f := ShallowCopy( e );
f[i] := j-1;
Add( t, f );
od;
od;
exp := t;
od;
return exp;
end );
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