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