44.10 Quotient Methods

gap> f:=FreeGroup(2);;fp:=f/[f.1^6,f.2^6,(f.1*f.2)^12];
<fp group on the generators [ f1, f2 ]>
gap> hom:=MaximalAbelianQuotient(fp);
[ f1, f2 ] -> [ f1, f3 ]
gap> Size(Image(hom));
36

computes an epimorphism from the finitely presented group fpgrp to the largest p-group of class cl which is a quotient of fpgrp. If cl is omitted, the largest finite p-group quotient is determined.

gap> hom:=EpimorphismPGroup(fp,2);  
[ f1, f2 ] -> [ a1, a2 ]
gap> Size(Image(hom));
8
gap> hom:=EpimorphismPGroup(fp,3,7);         
[ f1, f2 ] -> [ a1, a2 ]
gap> Size(Image(hom));
6561

returns an epimorphism on the class n finite nilpotent quotient of the finitely presented group fpgrp. If n is omitted, the largest finite nilpotent quotient is taken.

gap> hom:=EpimorphismNilpotentQuotient(fp,7);
[ f1, f2 ] -> [ f1*f4, f2*f5 ]
gap> Size(Image(hom));
52488

A related operation which is also applicable to finitely presented groups is GQuotients, which computes all epimorphisms from a (finitely presented) group F onto a given (finite) group G, see GQuotients.

gap> GQuotients(fp,Group((1,2,3),(1,2)));
[ [ f1, f2 ] -> [ (1,2,3), (2,3) ],[ f1, f2 ] -> [ (2,3), (1,2,3) ],
  [ f1, f2 ] -> [ (1,3), (2,3) ] ]

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