Dear GAP Forum,
Kurt Ewald gives the following example
kurt_ewald@compuserve.com said: > G:=Group([ (5,7)(6,8), (1,2), (1,5)(2,6)(3,7)(4,8), > (1,3)(2,4)(5,7)(6,8), > (3,4)(5,6), (1,2)(3,4), (1,2)(3,4)(5,6)(7,8) ]) Z0=1; gap> > Z1:=Centre(G); Group([ ( 1, 2)( 3, 4)( 5, 6)( 7, 8) ]) gap> > hom:=NaturalHomomorphismByNormalSubgroup(G,Z1); Pcgs([ > (1,5)(2,6)(3,7)(4,8), (7,8), ( 3, 4), (1,3)(2,4), (5,8)(6,7), > (5,7)(6,8) > ]) -> [ f1, f2, f3, f4, f5, f6 ] gap> F:=FactorGroup(G,Z1); <pc group > of size 64 with 6 generators> gap> C:=Centre(F); Group([ f5*f6 ])
and then asks:
But how can I take the inverse Image of C ?
Perhaps the simplest way is
gap> Z2 := PreImages(hom,C); Group([ ( 1, 2)( 3, 4)( 5, 6)( 7, 8), (5,6)(7,8) ])
an alternative, using InverseGeneralMapping is:
gap> ihom := InverseGeneralMapping(hom); [ f1, f2, f3, f4, f5, f6, <identity> of ... ] -> [ (1,5)(2,6)(3,7)(4,8), (7,8), ( 3, 4), (1,3)(2,4), (5,8)(6,7), (5,7)(6,8), ( 1, 2)( 3, 4)( 5, 6)( 7, 8) ] gap> Images(ihom,C); Group([ ( 1, 2)( 3, 4)( 5, 6)( 7, 8), (5,6)(7,8) ])
Steve Linton