23.10 Group Presentations

In section Presentation Records we have described the funtion PresentationFpGroup which supplies a presentation record for a finitely presented group. The following function can be used to compute a presentation record for a concrete (e.,g. permutation or matrix) group.

PresentationViaCosetTable( G ) PresentationViaCosetTable( G, F, words )

PresentationViaCosetTable constructs a presentation record for the given group G. The method being used is John Cannon's relations finding algorithm which has been described in Can73 or in Neu82.

In its first form, if only the group G has been specified, it applies Cannon's single stage algorithm which, by plain element multiplication, computes a coset table of G with respect to its trivial subgroup and then uses coset enumeration methods to find a defining set of relators for G.

    gap> G := GeneralLinearGroup( 2, 7 );
    GL(2,7)
    gap> G.generators;
    [ [ [ Z(7), 0*Z(7) ], [ 0*Z(7), Z(7)^0 ] ],
      [ [ Z(7)^3, Z(7)^0 ], [ Z(7)^3, 0*Z(7) ] ] ]
    gap> Size( G );
    2016
    gap> P := PresentationViaCosetTable( G );
    << presentation with 2 gens and 5 rels of total length 46 >>
    gap> TzPrintRelators( P );
    #I  1. f.2^3
    #I  2. f.1^6
    #I  3. f.1*f.2*f.1*f.2*f.1*f.2*f.1*f.2*f.1*f.2*f.1*f.2
    #I  4. f.1*f.2*f.1^-1*f.2*f.1*f.2^-1*f.1^-1*f.2*f.1*f.2*f.1^-1*f.2^-1
    #I  5. f.1^2*f.2*f.1*f.2*f.1*f.2^-1*f.1^-1*f.2^-1*f.1^3*f.2^-1 

The second form allows to call Cannon's two stage algorithm which first applies the single stage algorithm to an appropriate subgroup H of G and then uses the resulting relators of H and a coset table of G with respect to H to find relators of G. In this case the second argument, F, is assumed to be a free group with the same number of generators as G, and words is expected to be a list of words in the generators of F which, when being evaluated in the corresponding generators of G, provide subgroup generators for H.

    gap> M12 := MathieuGroup( 12 );;
    gap> M12.generators;
    [ ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11), ( 3, 7,11, 8)( 4,10, 5, 6),
      ( 1,12)( 2,11)( 3, 6)( 4, 8)( 5, 9)( 7,10) ]
    gap> F := FreeGroup( "a", "b", "c" );
    Group( a, b, c )
    gap> words := [ F.1, F.2 ];
    [ a, b ]
    gap> P := PresentationViaCosetTable( M12, F, words );
    << presentation with 3 gens and 10 rels of total length 97 >>
    gap> G := FpGroupPresentation( P );
    Group( a, b, c )
    gap> G.relators;
    [ c^2, b^4, a*c*a*c*a*c, a*b^-2*a*b^-2*a*b^-2, a^11,
      a^2*b*a^-2*b^-2*a*b^-1*a^2*b^-1,
      a*b*a^-1*b*a^-1*b^-1*a*b*a^-1*b*a^-1*b^-1,
      a^2*b*a^2*b^-2*a^-1*b*a^-1*b^-1*a^-1*b^-1,
      a^2*b^-1*a^-1*b^-1*a*c*b*c*a*b*a*b, a^3*b*a^2*b*a^-2*c*a*b*a^-1*c*a
     ] 

Before it is returned, the resulting presentation is being simplified by Tietze Transformations), but without allowing it to eliminate any generators. This restriction guarantees that we get a bijection between the list of generators of G and the list of generators in the presentation. Hence, if the generators of G are redundant and if you don't care for the bijection, it may be convenient to apply the function SimplifyPresentation again.

    gap> H := Group(
    >  [ (2,5,3), (2,7,5), (1,8,4), (1,8,6), (4,8,6), (3,5,7) ], () );;
    gap> P := PresentationViaCosetTable( H );
    << presentation with 6 gens and 12 rels of total length 42 >>
    gap> SimplifyPresentation( P );
    #I  there are 4 generators and 10 relators of total length 36 

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