44.3 Preimages in the Free Group

returns the underlying free group for the finitely presented group G. This is the group generated by the free generators provided by FreeGeneratorsOfFpGroup(G).

FreeGeneratorsOfFpGroup returns the underlying free generators corresponding to the generators of the finitely presented group G which must be a full fp group.

FreeGeneratorsOfWholeGroup also works for subgroups of an fp group and returns the free generators of the full group that defines the family.

returns the relators of the finitely presented group G as words in the free generators provided by FreeGeneratorsOfFpGroup(G).

gap> f := FreeGroup( "a", "b" );;
gap> g := f / [ f.1^5, f.2^2, f.1^f.2*f.1 ];
<fp group on the generators [ a, b ]>
gap> Size( g );
10
gap> FreeGroupOfFpGroup( g ) = f;
true
gap> FreeGeneratorsOfFpGroup( g );
[ a, b ]
gap> RelatorsOfFpGroup( g );
[ a^5, b^2, b^-1*a*b*a ]

Elements of a finitely presented group are not words, but are represented using a word from the free group as representative. The following two commands obtain this representative, respectively create an element in the finitely presented group.

Let elm be an element of a group whose elements are represented as words with further properties. Then UnderlyingElement returns the word from the free group that is used as a representative for elm.

gap> w := g.1*g.2;
a*b
gap> IsWord( w );
false
gap> ue := UnderlyingElement( w );
a*b
gap> IsWord( ue );
true

If fam is the elements family of a finitely presented group and word is a word in the free generators underlying this finitely presented group, this operation creates the element with the representative word in the free group.

gap> ge := ElementOfFpGroup( FamilyObj( g.1 ), f.1*f.2 );
a*b
gap> ge in f;
false
gap> ge in g;
true

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