ApCoCoA-1:Symbolic data Computations

Computation of Non-abelian Groups

Baumslag groups

Baumslag-Solitar groups have the following presentation.

BS(m,n)<a, b | b*a^m = a^n*b> where m, n are natural numbers

XML data:

<vars>a[1],a[2],b[1],b[2]</vars>
<params>m,n</params>
<rels>
  <ncpoly>a[1]*a[2]-1</ncpoly>
  <ncpoly>a[2]*a[1]-1</ncpoly>
  <ncpoly>b[1]*b[2]-1</ncpoly>
  <ncpoly>b[2]*b[1]-1</ncpoly>
  <ncpoly>b[1]*a[1]^{m}-a[1]^{n}*b[1]</ncpoly>
</rels> 

We enumerate partial Groebner bases for the Baumslag-Solitar groups as follows.

/*Use the ApCoCoA package ncpoly.*/
Use ZZ/(2)[a[1..2],b[1..2]];
NC.SetOrdering("LLEX");
A1:=[[a[1],a[2]],[1]];
A2:=[[a[2],a[1]],[1]];
B1:=[[b[1],b[2]],[1]];
B2:=[[b[2],b[1]],[1]];
-- Relation ba^2=a^3b. Change 2 and 3 in "()" to make another relation
R:=[[b[1],a[1]^(2)],[a[1]^(3),b[1]]];
G:=[A1,A2,B1,B2,R];
-- Enumerate a partial Groebner basis (see NC.GB for more details)
NC.GB(G,31,1,100,1000);

/*Use the ApCoCoA package gbmr.*/
-- See NCo.BGB for more details on the parameters DB, LB and OFlag.
Define BS(M,N,DB,LB,OFlag)
  $apcocoa/gbmr.SetX("aAbB");
  $apcocoa/gbmr.SetOrdering("LLEX");
  G:= [["aA",""],["Aa",""],["bB",""],["bB",""]];
  BA:= "b";
  AB:= "b";
  For I:= 1 To ARGV[1] Do
    BA:= BA + "a"; 
  EndFor;
  For I:= 1 To ARGV[2] Do
    AB:= "a" + Ab; 
  EndFor;
  Append(G,[BA,AB]);
  Return $apcocoa/gbmr.BGB(G,DB,LB,OFlag);
EndDefine;

Dicyclic groups

Dicyclic groups have the following presentation.

Dic(n) = <a,b | a^{2n} = 1, a^n = b^2, bab^{-1} = a^{-1}>

XML data:

<vars>a,b[1],b[2]</vars>
<params>n</params>
<rels>
  <ncpoly>a^{2n}-1</ncpoly>
  <ncpoly>b[1]*b[2]-1</ncpoly>
  <ncpoly>b[2]*b[1]-1</ncpoly>
  <ncpoly>a^{n}-b[1]*b[1]</ncpoly>
  <ncpoly>b[1]*a*b[2]-a^{2n-1}</ncpoly>
</rels> 

Dihedral groups

Dihedral groups have the following presentation.

Dih(n) = <r,s | r^n = s^2 = (rs)^2 = 1>

XML data:

<vars>r,s</vars>
<params>n</params>
<rels>
  <ncpoly>r^{n}-1</ncpoly>
  <ncpoly>s*s-1</ncpoly>
  <ncpoly>r*s*r*s-1</ncpoly>
</rels> 

von Dyck groups

Von Dyck groups have the following presentation.

D(l,m,n) = <x,y | x^{l} = y^{m} = (xy)^{n} = 1>

XML data:

<vars>x,y</vars>
<params>l,m,n</params>
<rels>
  <ncpoly>x^{l}-1</ncpoly>
  <ncpoly>y^{m}-1</ncpoly>
  <ncpoly>(x*y)^{n}-1</ncpoly>
</rels> 

Higman group

The Higman group has the following presentation.

H = <a,b,c,d | a^{-1}ba = b^2, b^{-1}cb = c^2, c^{-1}dc = d^2, d^{-1}ad = a^2>

XML data:

<vars>a[1],a[2],b[1],b[2],c[1],c[2],d[1],d[2]</vars>
<rels>
  <ncpoly>a[1]*a[2]-1</ncpoly>
  <ncpoly>a[2]*a[1]-1</ncpoly>
  <ncpoly>b[1]*b[2]-1</ncpoly>
  <ncpoly>b[2]*b[1]-1</ncpoly>
  <ncpoly>c[1]*c[2]-1</ncpoly>
  <ncpoly>c[2]*c[1]-1</ncpoly>
  <ncpoly>d[1]*d[2]-1</ncpoly>
  <ncpoly>d[2]*d[1]-1</ncpoly>
  <ncpoly>a[2]*b[1]*a[1]-b[1]*b[1]</ncpoly>
  <ncpoly>b[2]*c[1]*b[1]-c[1]*c[1]</ncpoly>
  <ncpoly>c[2]*d[1]*c[1]-d[1]*d[1]</ncpoly>
  <ncpoly>d[2]*a[1]*d[1]-a[1]*a[1]</ncpoly>
</rels> 

Ordinary tetrahedon groups

Ordinary tetrahedon groups have the following presentation where e_i >= 2 and f_i >= 2 for all i.

G(e_1,e_2,e_3,f_1,f_2,f_3) = <x,y,z | x^{e_1} = y^{e_2} = z^{e_3} = (xy^{-1})^{f_1} = (yz^{-1})^{f_2} = (zx^{-1})^{f_3} = 1>

XML data:

<vars>x,y,z</vars>
<params>e_1,e_2,e_3,f_1,f_2,f_3</params>
<rels>
  <ncpoly>x^{e_1}-1</ncpoly>
  <ncpoly>y^{e_2}-1</ncpoly>
  <ncpoly>z^{e_3}-1</ncpoly>
  <ncpoly>(x*y^{e_2-1})^{f_1}-1</ncpoly>
  <ncpoly>(y*z^{e_3-1})^{f_2}-1</ncpoly>
  <ncpoly>(z*x^{e_1-1})^{f_3}-1</ncpoly>
</rels> 

Thompson group

The Thompson group has a presentation as follows.

T = <a,b | [ab^{-1},a^{-1}ba] = [ab^{-1},a^{-2}ba^2] = 1>

XML data:

<vars>a[1],a[2],b[1],b[2]</vars>
<rels>
  <ncpoly>a[1]*a[2]-1</ncpoly>
  <ncpoly>a[2]*a[1]-1</ncpoly>
  <ncpoly>b[1]*b[2]-1</ncpoly>
  <ncpoly>b[2]*b[1]-1</ncpoly>
  <ncpoly>a[1]*b[2]*a[2]*b[1]*a[1]-b[1]*a[2]*a[2]*b[2]*a[1]</ncpoly>
  <ncpoly>a[1]*b[2]*a[2]*a[2]*b[1]*a[1]*a[1]-b[1]*a[2]*a[2]*a[2]*b[2]*a[1]*a[1]</ncpoly>
</rels> 

Triangle groups

Triangle groups have the following presentation.

Triangle(l,m,n) = {a,b,c | a^2 = b^2 = c^2 = (ab)^l = (bc)^m = (ca)^n = 1}

XML data:

<vars>a,b,c</vars>
<params>l,m,n</params>
<rels>
  <ncpoly>a*a-1</ncpoly>
  <ncpoly>b*b-1</ncpoly>
  <ncpoly>c*c-1</ncpoly>
  <ncpoly>(a*b)^{l}-1</ncpoly>
  <ncpoly>(b*c)^{m}-1</ncpoly>
  <ncpoly>(c*a)^{n}-1</ncpoly>
</rels> 

SL(3,8)

/* SL(3,8) has a presentation with generators a, b, c, d, e and the following relators
    a^2, b^2, c^7, de, ed,
    (cb)^2,
    (ba)^3,
    (acac^6)^2,
    c^2ac^6ac^5ac,
    dbe^2, 
    (ce)^2cd^2
    aead(ae)^2babd^2,
    eadae^2babd^2(ae)^2babd^2,
    ec^6daecdc^6aca,
    ec^6daecec^6d^2ae^2cd^2,
    ec^6daecec^6d^2ae^2cd^2.
    The following commands check whether the last relator, i.e. ec^6daecec^6d^2ae^2cd^2
    can be rewritten by the others, via Groebner basis techniques.
*/
Use ZZ/(2)[a,b,c,d,e];
G:=[[[a^2], [1]],
    [[b^2], [1]], 
    [[c^7], [1]],
    [[d, e], [1]],
    [[e, d], [1]],
    [[c, b, c, b], [1]],
    [[b, a, b, a, b, a], [1]],
    [[a, c, a, c^6, a, c, a, c^6], [1]],
    [[c^2, a, c^6, a, c^5, a, c], [1]],
    [[b, d, b, e^2], [1]], 
    [[c, e, c, e, c, d^2], [1]],
    [[a, e, a, d, a, e, a, e, b, a, b, d^2], [1]],
    [[e, a, d, a, e^2, b, a, b, d^2, a, e, a, e, b, a, b, d^2], [1]],
    [[e, c^6, d, a, e, c, d, c^6, a, c, a], [1]]];
F:=[ [e,c^6, d, a, e, c, e, c^6, d^2, a, e^2, c, d^2] ];
Gb:=NC.GB(G,31,1,100,5000);
NC.NR(F,Gb);