A finitely presented group is a group generated by a finite set of abstract generators subject only to a finite set of relations that these generators must satisfy.
A finitely presented group can be constructed in GAP as described in the chapter Finitely Presented Groups of the GAP gpl, see [GAP]. Here an alternative method of construction is described, closer to notation used for presentations in [PS97] and in [ATLAS]. One can construct a finitely presented group via a Coxeter graph which introduces generators and describes relations. If necessary extra generators or relations can be added. In this chapter procedures are described that create finitely presented groups via strings representing graphs, explicit generators and relations. Two procedures to form such a string presentation from an arbitrary finitely presented group are added as well.
For example from a string presentation to a finitely presented group:
gap> cox := "A3b5c3d3e4c"; "A3b5c3d3e4c" gap> rel := "A=(ce)^2"; "A=(ce)^2" gap> extrarel := "_PCE,(Abc)^5,(bcd)^5"; "_PCE,(Abc)^5,(bcd)^5"; gap> m11 := FpGroupViaStrings( cox, rel, extrarel ); && The map is: "Abcde" Group( f.1, f.2, f.3, f.4, f.5 ) gap> map := "Abcde"; gap> m10 := SubgroupViaStrings( m11, "b,c,d,AbcbAe", map ); Subgroup( Group( f.1, f.2, f.3, f.4, f.5 ), [ f.2, f.3, f.4, f.1*f.2*f.3*f.2*f.1*f.5 ] ) gap> Index(m11,m10); 11
From a finitely presented group to a string presentation:
gap> F5 := FreeGroup( "x1", "x2", "x3", "x4", "x5" );; gap> E1 := F5 / [ F5.1^2, F5.2^2, F5.3^2, F5.4^2, F5.5^2, > ( F5.1 * F5.3 )^2, ( F5.2 * F5.4 )^2, ( F5.1 * F5.2 )^3, > ( F5.2 * F5.3 )^3, ( F5.3 * F5.4 )^3, ( F5.4 * F5.1 )^3, > ( F5.1 * F5.5 )^3, ( F5.2 * F5.5 )^2, ( F5.3 * F5.5 )^3, > ( F5.4 * F5.5 )^2, > ( F5.1 * F5.2 * F5.3 * F5.4 * F5.3 * F5.2 )^2 ];; gap> CoxeterStringPresentation( E1 ); [ "A3B3C3D3A3E3C",, "(ABCDCB)^2" ]
The first section in this chapter describes Coxeter graphs. The next sections describe how to construct finitely presented groups and subgroups using strings representing Coxeter graphs, generators and words. The final sections describe procedures to create string presentations from arbitrary finitely presented groups.