Dear GAP-Forum,

The homomorphism routines for AgGroups contain an error that occurs when
computing the (nontrivial) kernel of an homomorphism into an AgGroup:

gap> g:=SymmetricGroup(3);;
gap> h:=TrivialSubgroup(SolvableGroup(2,1));;
gap> hom:=GroupHomomorphismByImages(g,h,g.generators,
     List(g.generators,i->h.identity));;
GroupHomomorphismByImages( Group( (1,3), (2,3) ), Subgroup( c2, [  ] ),
[ (1,3), (2,3) ], [ IdAgWord, IdAgWord ] )
gap> Kernel(hom);
Subgroup( Group( (1,3), (2,3) ), [  ] )

Reason for this error is a bad trick the `Kernel` routine plays on
`GroupHomomorphismByImages`: The inverse Mapping is constructed which is not
a true Homomorphism but only used for intermediate computations. This
mapping however is mangled by the AgGroup routines when trying to compute
a Cgs for the source subgroup.
Replacing the routine `AgGroupOps.GroupHomomorphismByImages` in the library
file `aghomom.g` by the following code bypasses the error:

#############################################################################
##
#F  AgGroupOps.GroupHomomorphismByImages( <U>, <R>, <g>, <i> ) . . create hom
##
AgGroupOps.GroupHomomorphismByImages := function( D, R, gens, imgs )

local   h,      # homomorphism
        i,
        ogens,
        oimgs,
        tmp;    # temporary

# Normalize <gens> and unbind possible '<D>.field'.
ogens := gens;
oimgs := imgs;
D := Normalized( D );
Unbind( D.field );
if Cgs(D) <> gens  then
    tmp  := AbstractIgs( D, gens, imgs );
    gens := tmp.igs;
    imgs := tmp.abstractIgs;
fi;

# If range <R> is just 'rec()', try to construct the image.
if not IsBound( R.generators )  then
    if 0 = Length(imgs)  then
        Error( "needs either range or at least one image" );
    fi;
    R := Group( imgs, imgs[1]^0 );
fi;

# Construct the homorphism record.
h := rec( source           := D,
          range            := R,
          domain           := Mappings,
          generators       := gens,
          genimages        := imgs,
          preimage         := D,
          image            := R.operations.Subgroup( Parent(R), imgs ),
          isGeneralMapping := true,
          operations       := AgGroupHomomorphismByImagesOps );

# check at least the given generator images <oimgs>
if not IsFpGroup(R)  then
    for i  in [ 1 .. Length(ogens) ]  do
        if Image( h, ogens[i] ) <> oimgs[i]  then
            return GroupOps.GroupHomomorphismByImages(D,R,ogens,oimgs);
        fi;
    od;
fi;
return h;

end;
AgGroupHomomorphismByImagesOps.Kernel :=
AgGroupHomomorphismByImagesOps.KernelGroupHomomorphism;

I apologize for any inconvenience this error might have caused.

Alexander Hulpke