73.142 FpPair

FpPair( G )

When G is a finitely presented group, this function finds a faithful permutation representation P, which may be the regular representation, and sets up a pairing between G and P.

    gap> f := FreeGroup( 2 );;
    gap> rels := [ f.1^3, f.2^3, (f.1*f.2)^2 ];;
    gap> g := f/rels;;
    gap> pairg := FpPair( g );
    rec(
      perm := Group( (2,4,3), (1,3,2) ),
      fp := Group( f.1, f.2 ),
      f2p := GroupHomomorphismByImages( Group( f.1, f.2 ), 
        Group( (2,4,3), (1,3,2) ), [ f.1, f.2 ], [ (2,4,3), (1,3,2) ] ),
      p2f := GroupHomomorphismByImages( Group( (2,4,3), (1,3,2) ),
        Group( f.1, f.2 ), [ (2,4,3), (1,3,2) ], [ f.1, f.2 ] ),
      isFpPair := true,
      isMinTransitivePair := true,
      generators := [ (2,4,3), (1,3,2) ],
      degree := 4,
      position := 3 )  

When G is a permutation group, the function PresentationViaCosetTable is called to find a presentation for G and hence a finitely presented group F isomorphic to G. When G has a name, the name <name of G>Fp is given automatically to F and <name of G>Pair to the pair.

    gap> h20.generators;
    [ (1,2,3,4,5), (2,3,5,4) ]
    gap> pairh := FpPair( h20 );
    rec(
      perm := h20,
      fp := h20Fp,
      f2p := GroupHomomorphismByImages( h20Fp, h20, [ f.1, f.2 ], 
        [ (1,2,3,4,5), (2,3,5,4) ] ),
      p2f := GroupHomomorphismByImages( h20, h20Fp,
        [ (1,2,3,4,5), (2,3,5,4) ], [ f.1, f.2 ] ),
      isFpPair := true,
      degree := 5,
      presentation := << presentation with 2 gens and 3 rels
        of total length 14 >>,
      name := [ 'h', '2', '0', 'P', 'a', 'i', 'r' ] )
    gap> pairh.fp.relators;
    [ f.2^4, f.1^5, f.1*f.2*f.1*f.2^-1*f.1 ]   

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