5 Subgroups of L-presented groups

creates the subgroup U of G generated by gens. The Parent value of U will be G.

For example, the branching subgroup of the Grigorchuk group can be defined as follows

gap> G := ExamplesOfLPresentations(1);;
gap> a := G.1;; b := G.2;; c := G.3;; d := G.4;;
gap> K := Subgroup( G, [ Comm( a, b ), Comm( b^a, d ), Comm( b, d^a ) ] );
Group([ a^-1*b^-1*a*b, b^-1*a^-1*d^-1*a*b*a^-1*d*a, a^-1*b^-1*a*d^-1*a^-1*b*a*d ])

creates the subgroup U of G which is represented by the coset-table Tab.

For instance, the branching subgroup of the Grigorchuk group can be defined by the following coset-table

gap> Tab := [ [ 2, 1, 6, 9, 10, 3, 11, 12, 4, 5, 7, 8, 15, 16, 13, 14 ],
  [ 2, 1, 6, 9, 10, 3, 11, 12, 4, 5, 7, 8, 15, 16, 13, 14 ],
  [ 3, 6, 1, 5, 4, 2, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15 ],
  [ 3, 6, 1, 5, 4, 2, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15 ],
  [ 4, 7, 5, 1, 3, 8, 2, 6, 13, 14, 15, 16, 9, 10, 11, 12 ],
  [ 4, 7, 5, 1, 3, 8, 2, 6, 13, 14, 15, 16, 9, 10, 11, 12 ],
  [ 5, 8, 4, 3, 1, 7, 6, 2, 14, 13, 16, 15, 10, 9, 12, 11 ],
  [ 5, 8, 4, 3, 1, 7, 6, 2, 14, 13, 16, 15, 10, 9, 12, 11 ] ]
gap> U := SubgroupLpGroupByCosetTable( G, Tab );
Group(<subgroup of L-presented group, no generators known>)
gap> U = K;
true

The generators of U can be computed with the Schreier-algorithm which is implemented in the method 'GeneratorsOfGroup'.

5.2 Computing the index of finite index subgroups

In principle, it is possible to compute the index of a finite index subgroup of an LpGroup Har10. The method reduces the case to certain finitely presented groups by applying only finitely many endomorphisms to the iterated relations. It then uses coset enumeration for finitely presented groups to compute an upper bound on the index of the subgroup. If the coset enumeration for finitely presented groups terminated, the method attempts to prove that the upper bound is sharp. For further details we refer to Har10.

attempts to compute the index of H in its parent group.

gap> G:=ExamplesOfLPresentations(1);;
gap> a := G.1;; b := G.2;; c := G.3;; d := G.4;;
gap> K := Subgroup( G, [ Comm(a,b), Comm( b, d^a ), Comm( b^a, d )] );;
gap> IndexInWholeGroup( K );
16

attempts to compute the index of I in the subgroup H. The subgroup I must be contained in H.

gap> G:=ExamplesOfLPresentations(1);;
gap> a := G.1;; b := G.2;; c := G.3;; d := G.4;;
gap> K := Subgroup( G, [ Comm(a,b), Comm( b, d^a ), Comm( b^a, d )] );;
gap> KxK := Subgroup( G, [ Comm(b,d^a), Comm(b^a,d), Comm(d^a,c^(a*c)),                         
> Comm( d^(a*c), c^a), Comm( d, c^(a*c*a) ), Comm( d^(a*c*a), c) ] );;
gap> Index( K, KxK );
4

computes a coset-table for the subgroup H in its parent group.

gap> CosetTableInWholeGroup( K );
[ [ 2, 1, 6, 9, 10, 3, 11, 12, 4, 5, 7, 8, 15, 16, 13, 14 ],
  [ 2, 1, 6, 9, 10, 3, 11, 12, 4, 5, 7, 8, 15, 16, 13, 14 ],
  [ 3, 6, 1, 5, 4, 2, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15 ],
  [ 3, 6, 1, 5, 4, 2, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15 ],
  [ 4, 7, 5, 1, 3, 8, 2, 6, 13, 14, 15, 16, 9, 10, 11, 12 ],
  [ 4, 7, 5, 1, 3, 8, 2, 6, 13, 14, 15, 16, 9, 10, 11, 12 ],
  [ 5, 8, 4, 3, 1, 7, 6, 2, 14, 13, 16, 15, 10, 9, 12, 11 ],
  [ 5, 8, 4, 3, 1, 7, 6, 2, 14, 13, 16, 15, 10, 9, 12, 11 ] ]

5.3 Technical details

For performance issues the following global variables can be used to modify the behaviour of the coset enumeration:

defines the maximal word-length of endomorphisms in the free monoid which are applied to the iterated relations.

defines the coset enumeration process used for finitely presented groups. It should be a function which take as input a subgroup h of a finitely presented group and it computes a coset table in the whole group. The default uses the following method of the ACE-package

function ( h )
    local  f, rels, gens;
    f := FreeGeneratorsOfFpGroup( Parent( h ) );
    rels := RelatorsOfFpGroup( Parent( h ) );
    gens := List( GeneratorsOfGroup( h ), UnderlyingElement );
    return ACECosetTable( f, rels, gens : silent := true,
        hard := true,
        max := 10 ^ 8,
        Wo := 10 ^ 8 );

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