CoCalc -- triangulaire.sagews

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### Statistiques sur les quotients finis de groupes
### donnés par une présentation triangulaire

## voir les tables à la fin de l'article
#
#    Donald I. Cartwright; Anna Maria Mantero; Tim Steger; Anna Zappa
#    Groups acting simply transitively on the vertices of a building of type A~2. II. The cases q=2 and q=3.
#    Geom. Dedicata 47 (1993), no. 2, 167–223.
#    20E42 (20G25 51E24)
#    http://www.ams.org/mathscinet-getitem?mr=1232966

︠237c7dc0-bf3a-4623-8d7a-60bc15d2e972︠

# http://doc.sagemath.org/html/en/reference/groups/sage/groups/finitely_presented.html
# http://www.gap-system.org/Manuals/doc/ref/chap47.html
# http://mail.gap-system.org/pipermail/forum/2003/000656.html

# ruse pour afficher la commande gap à copier-coller

for i in xrange(7):
    i, j, k = i+1, (i+1)%7 + 1, (i+3)%7 + 1
    print 'f.' + str(i) + ' * f.' + str(j) + ' * f.' + str(k) + ','

f.1 * f.2 * f.4,
f.2 * f.3 * f.5,
f.3 * f.4 * f.6,
f.4 * f.5 * f.7,
f.5 * f.6 * f.1,
f.6 * f.7 * f.2,
f.7 * f.1 * f.3,

# ruse pour afficher la commande gap à copier-coller

for i in xrange(7):
    i = i + 1
    print 'a' + str(i) + " := g." + str(i) + ';;'

a1 := g.1;;
a2 := g.2;;
a3 := g.3;;
a4 := g.4;;
a5 := g.5;;
a6 := g.6;;
a7 := g.7;;

# creons en GAP un groupe libre puis son quotient

%gap
f := FreeGroup("a1", "a2", "a3", "a4", "a5", "a6", "a7");;

g := f / [ f.1 * f.2 * f.4,
f.2 * f.3 * f.5,
f.3 * f.4 * f.6,
f.4 * f.5 * f.7,
f.5 * f.6 * f.1,
f.6 * f.7 * f.2,
f.7 * f.1 * f.3 ] ;

%gap
g;

<fp group on the generators [ a1, a2, a3, a4, a5, a6, a7 ]>

%gap
GeneratorsOfGroup(g);

[ a1, a2, a3, a4, a5, a6, a7 ]

%gap
a1 := g.1;;
a2 := g.2;;
a3 := g.3;;
a4 := g.4;;
a5 := g.5;;
a6 := g.6;;
a7 := g.7;;

%gap
a3

a3

# On peut obtenir la liste des sous-groupes d'indice maxi k.
# Ci-dessous pour k = 12 (ça prend quelques minutes).

%gap
li := LowIndexSubgroupsFpGroup( g, TrivialSubgroup( g ), 12);

[ Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>), 
  Group(<fp, no generators known>), Group(<fp, no generators known>) ]

# indices des groupes de la liste

%gap
for h in li do
    Print(Index(g,h));
    Print(", ");
od;

1, 2, 2, 2, 4, 8, 8, 8, 7, 2, 2, 4, 8, 8, 8, 2, 2, 3, 6, 4, 8, 6, 6, 8, 12, 
8, 8, 4, 8, 8, 8, 6, 6, 4, 12, 8, 8, 8, 6, 6, 4, 8, 12, 8, 8, 12, 12, 4, 12, 
8, 8, 8, 12, 8,

# lesquels sont normaux?

%gap
for h in li do
    Print(Index(g,h));
    if IsNormal(g,h) then Print("n"); fi;
    Print(", ");
od;

1n, 2n, 2n, 2n, 4n, 8, 8, 8, 7, 2n, 2n, 4n, 8, 8, 8, 2n, 2n, 3n, 6n, 4n, 8, 
6n, 6n, 8, 12n, 8, 8, 4n, 8, 8, 8, 6n, 6n, 4n, 12n, 8, 8, 8, 6n, 6n, 4n, 8, 
12n, 8, 8, 12n, 12n, 4n, 12n, 8, 8, 8, 12n, 8n,

# pour les sous-groupes normaux on peut construire le quotient
# et décrire la structure de ce groupe quotient
#
# http://www.gap-system.org/Faq/faq.html#7.11

%gap
for h in li do
    if IsNormal(g,h) then
        nhom := NaturalHomomorphismByNormalSubgroup(g,h);
        gmodh := Image(nhom);
        Print(StructureDescription(gmodh));
        Print("\n");
    fi;
od;

1
C2
C2
C2
C2 x C2
C2
C2
C2 x C2
C2
C2
C3
C6
C2 x C2
C6
C6
C6 x C2
C2 x C2
C6
C6
C2 x C2
C6 x C2
C6
C6
C2 x C2
C6 x C2
C6 x C2
C6 x C2
C2 x C2
C6 x C2
C6 x C2
C2 x C2 x C2


#Maintenant avec la présentation numéro 9.1
%gap
fb := FreeGroup("a1", "a2", "a3", "a4", "a5", "a6", "a7", "a8", "a9", "a10", "a11", "a12", "a13");;

%gap
fb

<free group on the generators [ a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, 
  a12, a13 ]>

%gap
gb := fb / [
    fb.13 * fb.13 * fb.13,
    fb.4 * fb.4 * fb.4,
    fb.12 * fb.12 * fb.12,
    fb.10 * fb.10 * fb.10,
    fb.4 * fb.12 * fb.10,
    fb.13 * fb.7 * fb.3,
    fb.13 * fb.8 * fb.9,
    fb.13 * fb.11 * fb.1,
    fb.1 * fb.3 * fb.2,
    fb.3 * fb.9 * fb.6,
    fb.9 * fb.1 * fb.5,
    fb.1 * fb.12 * fb.6,
    fb.3 * fb.10 * fb.5,
    fb.9 * fb.4 * fb.2,
    fb.2 * fb.8 * fb.11,
    fb.6 * fb.11 * fb.7,
    fb.5 * fb.7 * fb.8,
    fb.2 * fb.12 * fb.7,
    fb.6 * fb.10 * fb.8,
    fb.5 * fb.4 * fb.11
    ] ;

%gap
gb;

<fp group on the generators [ a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, 
  a12, a13 ]>

# Finding low index subgroups up to index 6 takes a long time
%gap
li6 := LowIndexSubgroupsFpGroup( gb, TrivialSubgroup( gb ), 6);

[ Group(<fp, no generators known>), Group(<fp, no generators known>) ]


%gap
for h in li6 do
    Print(Index(gb,h));
    Print(", ");
od;

1, 3,

# Finding low index subgroups up to index 12 takes too long.
# We should try on sagange.

%gap
li12 := LowIndexSubgroupsFpGroup( gb, TrivialSubgroup( gb ), 12);


%gap
for h in li12 do
    Print(Index(gb,h));
    Print(", ");
od;


︠438c15cf-4aa5-4043-aadd-af6c2fb17f68︠
%gap
PQuotient(gb,2);

<2-quotient system of 2-class 1 with 0 generators>


%gap
hom:=EpimorphismPGroup(gb,2); Size(Image(hom));

[ a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13 ] -> 
[ <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ... ]
1

1

%gap
hom:=EpimorphismPGroup(g,2,10);Size(Image(hom));

[ a1, a2, a3, a4, a5, a6, a7 ] -> 
[ a1, a2, a3, a1*a2*a4*a5*a7*a13*a15*a16*a22*a24*a25, 
  a2*a3*a4*a6*a9*a10*a13*a21*a23*a24*a25*a26, 
  a1*a2*a3*a4*a5*a6*a7*a8*a10*a11*a12*a13*a14*a15*a17*a19*a20*a21*a23*a25, 
  a1*a3*a4*a8*a10*a11*a12*a14*a15*a16*a17*a19*a22*a23*a26*a27 ]
134217728


︠bcb4f46e-4eac-461c-9e85-f8680f86ad68︠
%gap
hom:=EpimorphismPGroup(gb,3); Size(Image(hom));

[ a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13 ] -> 
[ a1, a1, a1, a1, a1, a1, a1, a1, a1, a1, a1, a1, a1 ]
3


%gap
hom:=EpimorphismPGroup(gb,5); Size(Image(hom));

[ a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13 ] -> 
[ <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ... ]
1

%gap
hom:=EpimorphismPGroup(gb,7); Size(Image(hom));

[ a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13 ] -> 
[ <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ... ]
1


%gap
hom:=EpimorphismPGroup(g,3,100);Size(Image(hom));

[ a1, a2, a3, a4, a5, a6, a7 ] -> [ a1, a1, a1, a1, a1, a1, a1 ]
3

%gap
hom:=EpimorphismPGroup(gb,11); Size(Image(hom));

[ a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13 ] -> 
[ <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ... ]
1

%gap
hom:=EpimorphismPGroup(gb,13); Size(Image(hom));

[ a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13 ] -> 
[ <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ..., <identity> of ..., <identity> of ..., <identity> of ..., 
  <identity> of ... ]
1


#Witzel's presentation
%gap
fw:= FreeGroup("a1", "a2", "a3");;



%gap
gw := fw / [
    fw.1^7, fw.2^7,fw.3^7, fw.1*fw.2*fw.3^3, fw.1^3*fw.2^3*fw.3
    ] ;

<fp group on the generators [ a1, a2, a3 ]>


%gap
hom:=EpimorphismPGroup(gw,2);Size(Image(hom));

[ a1, a2, a3 ] -> [ <identity> of ..., <identity> of ..., <identity> of ... ]
1


%gap
hom:=EpimorphismPGroup(gw,3);Size(Image(hom));

[ a1, a2, a3 ] -> [ <identity> of ..., <identity> of ..., <identity> of ... ]
1



%gap
hom:=EpimorphismPGroup(gw,5);Size(Image(hom));

[ a1, a2, a3 ] -> [ <identity> of ..., <identity> of ..., <identity> of ... ]
1



%gap
hom:=EpimorphismPGroup(gw,7);Size(Image(hom));

[ a1, a2, a3 ] -> [ a1, a1^6, <identity> of ... ]
7


%gap
hom:=EpimorphismPGroup(gw,11);Size(Image(hom));

[ a1, a2, a3 ] -> [ <identity> of ..., <identity> of ..., <identity> of ... ]
1


%gap
hom:=EpimorphismPGroup(gw,13);Size(Image(hom));

[ a1, a2, a3 ] -> [ <identity> of ..., <identity> of ..., <identity> of ... ]
1


%gap
hom:=EpimorphismPGroup(gw,17);Size(Image(hom));

[ a1, a2, a3 ] -> [ <identity> of ..., <identity> of ..., <identity> of ... ]
1


%gap
ga := fw / [
    fw.1^7, fw.2^7,fw.3^7, fw.1*fw.2*fw.3, fw.1^3*fw.2^5*fw.3
    ] ;

<fp group on the generators [ a1, a2, a3 ]>



%gap
hom:=EpimorphismPGroup(ga,2);Size(Image(hom));

[ a1, a2, a3 ] -> [ <identity> of ..., <identity> of ..., <identity> of ... ]
1


%gap
hom:=EpimorphismPGroup(ga,3);Size(Image(hom));

[ a1, a2, a3 ] -> [ <identity> of ..., <identity> of ..., <identity> of ... ]
1

%gap
hom:=EpimorphismPGroup(ga,5);Size(Image(hom));

[ a1, a2, a3 ] -> [ <identity> of ..., <identity> of ..., <identity> of ... ]
1

%gap
hom:=EpimorphismPGroup(ga,7);Size(Image(hom));

[ a1, a2, a3 ] -> [ a1, a1^3, a1^3 ]
7


%gap
hom:=EpimorphismPGroup(ga,11);Size(Image(hom));

[ a1, a2, a3 ] -> [ <identity> of ..., <identity> of ..., <identity> of ... ]
1