wrong group order of a preimage of a group homomorphism

[ThomasBreuer]

The following happens in GAP 4.13.0 as well as in the current master branch.

gap> G:= AbelianGroup( IsPcGroup, [ 3, 3, 3 ] );;
gap> Size( G );  
27
gap> H:= AbelianGroup( IsPcGroup, [ 3, 3 ] );;
gap> f:= GroupHomomorphismByImages( G, H, GeneratorsOfGroup( G ),
>         [ GeneratorsOfGroup( H )[1], One(H), One(H) ] );;
gap> ker:= Kernel( f );;
gap> Size( ker );
9
gap> pre:= PreImage( f, H );;
gap> HasSize( pre );
true
gap> Size( pre );
81

When the input line Size( ker ) is left out then the HasSize output is false, and the final Size value is correct.

(I found this problem when working on oscar-system/Oscar.jl/pull/3166.)

[ThomasBreuer]

The reason for the above wrong result is the following.

gap> G:= AbelianGroup( IsPcGroup, [ 3, 3, 3 ] );;
gap> H:= AbelianGroup( IsPcGroup, [ 3, 3 ] );;
gap> f:= GroupHomomorphismByImages( G, H, GeneratorsOfGroup( G ),
>         [ GeneratorsOfGroup( H )[1], One(H), One(H) ] );;
gap> x:= GeneratorsOfGroup( H )[2];
f2
gap> x in Image( f );
false
gap> x in Range( f );
true
gap> PreImagesRepresentative( f, x );
<identity> of ...

According to the documentation of PreImagesRepresentative, the last result should be fail.

By the way:
If we reduce the example further, we find another bug:

gap> G:= AbelianGroup( IsPcGroup, [ 3, 3 ] );;
gap> H:= AbelianGroup( IsPcGroup, [ 3 ] );;
gap> f:= GroupHomomorphismByImages( G, H, GeneratorsOfGroup( G ),
>         [ One( H ), One(H) ] );;
gap> x:= GeneratorsOfGroup( H )[1];
f1
gap> x in Image( f );
false
gap> x in Range( f );                                                 
true
gap> PreImagesRepresentative( f, x );
Error, List Element: <list>[1] must have an assigned value in
[...]

(In fact two PreImagesRepresentative methods are applicable in the above situaton. Both run into an error.)

[fingolfin]

See also PR #5073 by @cdwensley who has been trying to improve the situations... We should join forces with him to make a stronger push on this...

[fingolfin]

And also see your own issue #4809

[fingolfin]

In this particular case I think the core of the issue is that ExponentsOfPcElement(pcgs,elm) simply returns garbage if elm is not in the group generated by pcgs

[fingolfin]

Well, that and that this method does not even have the possibility of returning fail.

InstallMethod( PreImagesRepresentative, "method for pcgs hom",
  FamRangeEqFamElm,
  [ IsToPcGroupHomomorphismByImages,IsMultiplicativeElementWithInverse ], 0,
function( hom, elm )
    local  pcgsR, exp, imgs, pre, i;

    # precompute pcgs
    InversePcgs( hom );

    pcgsR := hom!.rangePcgs;
    exp := ExponentsOfPcElement( pcgsR, elm );
    imgs := hom!.rangePcgsPreimages;
    pre := Identity( Source( hom ) );
    for i in [1..Length(exp)] do
      if exp[i]>0 then
        pre := pre * imgs[i]^exp[i];
      fi;
    od;
    return pre;
end);