23.12 SimplifiedFpGroup

SimplifiedFpGroup( G )

SimplifiedFpGroup applies Tietze transformations to a copy of the presentation of the given finitely presented group G in order to reduce it with respect to the number of generators, the number of relators, and the relator lengths.

SimplifiedFpGroup returns the resulting finitely presented group (which is isomorphic to G).

    gap> F6 := FreeGroup( 6, "G" );;
    gap> G := F6 / [ F6.1^2, F6.2^2, F6.4*F6.6^-1, F6.5^2, F6.6^2,
    >         F6.1*F6.2^-1*F6.3, F6.1*F6.5*F6.3^-1, F6.2*F6.4^-1*F6.3,
    >         F6.3*F6.4*F6.5^-1, F6.1*F6.6*F6.3^-2, F6.3^4 ];;
    gap> H := SimplifiedFpGroup( G );
    Group( G.1, G.3 )
    gap> H.relators;
    [ G.1^2, G.1*G.3^-1*G.1*G.3^-1, G.3^4 ] 

In fact, the command

H := SimplifiedFpGroup( G );

is an abbreviation of the command sequence

    P := PresentationFpGroup( G, 0 );;
    SimplifyPresentation( P );
    H := FpGroupPresentation( P ); 

which applies a rather simple-minded strategy of Tietze transformations to the intermediate presentation record P (see Presentation Records). If for some concrete group the resulting presentation is unsatisfying, then you should try a more sophisticated, interactive use of the available Tietze transformation commands (see Tietze Transformations).

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