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##
#W sta.gi GUARANA package Bjoern Assmann
##
## Examples that were used in the Groups St Andrews 2005 paper
## "Algorithmic use of the Mal'cev correspondence".
##
#H @(#)$Id$
##
#Y 2006
##
##
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##
#F GUARANA.GetSomeAutomorphsimOfF_nc( F_nc, n )
##
## IN
## F_nc ..... free nilpotent group on n gens and class c
## n ........ number of generators.
##
## OUT
## A list of autmorphsims of F_nc.
## Some of them are induced by automorphisms of F_n.
##
GUARANA.GetSomeAutomorphsimOfF_nc := function( F_nc, n )
local F,A,gensA,gensN,k,auts,ll,gensF,imgs,g,beta,i,aut;
# get automorphsim induced from free group
F := FreeGroup( n );
gensF := GeneratorsOfGroup( F );
A := AutomorphismGroup( F );
gensA := GeneratorsOfGroup( A );
gensN := Pcp( F_nc );
k := Length( gensA );
auts := [];
for i in [1..k] do
aut := gensA[i];
ll := List( gensF, x->x^aut );
imgs := List( ll, x -> MappedWord( x, gensF, gensN{[1..n]} ) );
Add( auts, GroupHomomorphismByImagesNC(F_nc,F_nc,gensN{[1..n]},imgs ));
od;
# get automorphism of Bryant's paper
# "Automorphism groups of free nilpotent groups"
# Arch. Math. Basel. 52, 1989.
g := gensN[1]*Comm( Comm( gensN[1], gensN[2] ), gensN[1] );
beta := GroupHomomorphismByImagesNC( F_nc,F_nc,gensN{[1,2]}, [g,gensN[2]]);
Add( auts, beta );
return auts;
end;
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##
#F GUARANA.PcpTGroupByAbelianGroup( N, auts )
##
## IN
## N ..... T-group given by a finite presentation
## auts... List Automorphisms of N, where <auts> is abelian
##
## OUT
## polycyclic presentation of the semidirect product H \rhd N
## where H is a free abelian group with Length(auts) generators, which
## acts like auts on N
##
GUARANA.PcpGroupTGroupByAbelianGroup := function( N, auts )
local n,m,l,coll,pcpN,exp1,i,j,con,exp,exp2,genList,g;
#get collector
n := HirschLength( N );
m := Length( auts );
l := m+n;
coll := FromTheLeftCollector( l );
# define conjugation relations within N, g_i^g_j
pcpN := Pcp( N );
exp1 := List( [1..m], x-> 0 );
for i in [2..n] do
for j in [1..(i-1)] do
# compute g_i^g_j
con := pcpN[i]^pcpN[j];
exp2 := Exponents( con );
exp := Concatenation( exp1, exp2 );
genList:=GUARANA.Exp2GenList(exp);
SetConjugate( coll, i+m, j+m, genList);
# compute g_i^(g_j^-1)
con := pcpN[i]^(pcpN[j]^-1);
exp2 := Exponents( con );
exp := Concatenation( exp1, exp2 );
genList:=GUARANA.Exp2GenList(exp);
SetConjugate( coll, i+m, -(j+m), genList);
od;
od;
#define relations coming from the action of H on N
for j in [1..m] do
for i in [1..n] do
#action of g_j on g_{i+m}
g := pcpN[i]^(auts[j]);
exp2 := Exponents( g );
exp := Concatenation( exp1, exp2 );
genList:=GUARANA.Exp2GenList(exp);
SetConjugate( coll, i+m, j, genList);
#action of g_j^-1 onf g_{i+m}
g := pcpN[i]^(auts[j]^-1);
exp2 := Exponents( g );
exp := Concatenation( exp1, exp2 );
genList:=GUARANA.Exp2GenList(exp);
SetConjugate( coll, i+m, -j, genList);
od;
od;
UpdatePolycyclicCollector( coll);
return PcpGroupByCollector( coll );
end;
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##
#F GUARANA.NilpotentByFreeAbelianExams( n )
##
## IN
## n ........... natural number between 1 and 8 for the
## Sta paper examples.
## Further choices [-11..-1]
## [10..29].
##
## OUT
## Pcp of a nilpotent by free abelian group.
##
GUARANA.NilpotentByFreeAbelianExams := function( n )
local x,pol,R,G,m,ll,N,n1,n2,N2,F,F_nc,hl,auts,T,a;
x := Indeterminate( Rationals );
# first 8 groups as in Groups St Andrews 2005 paper
if n=1 then
pol := x^2-3;
R := GUARANA.Triang_PresentTriang( 3, pol );
return GUARANA.Triang_UpperTriangAndUnitriang( R );
fi;
if n=2 then
pol := x^2-3;
R := GUARANA.Triang_PresentTriang( 4, pol );
return GUARANA.Triang_UpperTriangAndUnitriang( R );
fi;
if n=3 then
pol := x^2-3;
R := GUARANA.Triang_PresentTriang( 5, pol );
return GUARANA.Triang_UpperTriangAndUnitriang( R );
fi;
if n=4 then
pol := x^3 - x^2 + 4;
R := GUARANA.Triang_PresentTriang( 3, pol );
return GUARANA.Triang_UpperTriangAndUnitriang( R );
fi;
if n=5 then
pol := x^3 - x^2 + 4;
R := GUARANA.Triang_PresentTriang( 4, pol );
return GUARANA.Triang_UpperTriangAndUnitriang( R );
fi;
if n=6 then
m := 2;
F := FreeGroup( m );
F_nc := GUARANA.NilpotentQuotient( F, 4 );
hl := HirschLength( F_nc );
auts := GUARANA.GetSomeAutomorphsimOfF_nc( F_nc, m );
a := auts[1]*auts[2]*auts[3]^3;
G := GUARANA.PcpGroupTGroupByAbelianGroup( F_nc, [a] );
N := Subgroup( G, Pcp(G){[2..hl+1]} );
T := rec( G := G, N := N );
return T;
fi;
if n=7 then
m := 2;
F := FreeGroup( m );
F_nc := GUARANA.NilpotentQuotient( F, 5 );
hl := HirschLength( F_nc );
auts := GUARANA.GetSomeAutomorphsimOfF_nc( F_nc, m );
a := auts[1]*auts[2]*auts[3]^3;
G := GUARANA.PcpGroupTGroupByAbelianGroup( F_nc, [a] );
N := Subgroup( G, Pcp(G){[2..hl+1]} );
T := rec( G := G, N := N );
return T;
fi;
if n=8 then
m := 3;
F := FreeGroup( m );
F_nc := GUARANA.NilpotentQuotient( F, 4 );
hl := HirschLength( F_nc );
auts := GUARANA.GetSomeAutomorphsimOfF_nc( F_nc, m );
a := auts[1]*auts[3]^3;
G := GUARANA.PcpGroupTGroupByAbelianGroup( F_nc, [a] );
N := Subgroup( G, Pcp(G){[2..hl+1]} );
T := rec( G := G, N := N );
return T;
fi;
# other examples
if n=11 then
pol := x^3 - x^2 + 4;
R := GUARANA.Triang_PresentTriang( 4, pol );
return GUARANA.Triang_UpperTriangAndUnitriang( R );
fi;
if n=12 then
pol := x^3 - x^2 + 4;
R := GUARANA.Triang_PresentTriang( 3, pol );
G := R.Tr;
m := Length( Pcp(G) );
ll := [R.noD+1..m];
N := Subgroup( G, Pcp(G){ll} );
n1 := Pcp(N)[2]*Pcp(N)[4];
n2 := Pcp(N)[5]*Pcp(N)[1]^-1;
N2 := NormalClosure( G, Subgroup( G, [n1,n2]) );
return rec( G := G, N := N2 );
fi;
if n=13 then
pol := x^7-x^6+x^5-x^4-3*x^3+3*x^2-2*x+1;
R := GUARANA.Triang_PresentTriang( 3, pol );
return GUARANA.Triang_UpperTriangAndUnitriang( R );
fi;
if n=14 then
pol := x^3 - x^2 + 4;
R := GUARANA.Triang_PresentTriang( 4, pol );
return GUARANA.Triang_UpperTriangAndUnitriang( R );
fi;
if n=15 then
pol := x^3 - x^2 + 4;
R := GUARANA.Triang_PresentTriang( 4, pol );
G := R.Tr;
m := Length( Pcp(G) );
ll := [R.noD+1..m];
N := Subgroup( G, Pcp(G){ll} );
n1 := Pcp(N)[2]*Pcp(N)[5]*Pcp(N)[13]^-1;
n2 := Pcp(N)[3]*Pcp(N)[7]*Pcp(N)[11]^-1;
N2 := NormalClosure( G, Subgroup( G, [n1,n2]) );
return rec( G := G, N := N2 );
fi;
if n = 11 then
pol := x^2 -3;
R := GUARANA.Triang_PresentTriang( 5, pol );
return GUARANA.Triang_UpperTriangAndUnitriang( R );
fi;
if n = 21 then
m := 2;
F := FreeGroup( m );
F_nc := GUARANA.NilpotentQuotient( F, 5 );
hl := HirschLength( F_nc );
auts := GUARANA.GetSomeAutomorphsimOfF_nc( F_nc, 2 );
a := auts[1]*auts[2]*auts[3]^3;
G := GUARANA.PcpGroupTGroupByAbelianGroup( F_nc, [a] );
N := Subgroup( G, Pcp(G){[2..hl+1]} );
T := rec( G := G, N := N );
return T;
fi;
if n = 29 then
# the calculation of the presentation needs 3000 sec.
# Dimension = 34
F := FreeGroup( 2 );
return GUARANA.NilpotentQuotient( F, 6 );
fi;
if n in [-10..-1] then
G := ExamplesOfSomePcpGroups(-n);
N := FittingSubgroup( G );
return rec( G := G, N := N );
fi;
if n = -11 then
G := ExamplesOfSomePcpGroups(13);
N := FittingSubgroup( G );
return rec( G := G, N := N );
fi;
end;
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##
#E
