Let
$G$
be any group, and
$H$
be a normal subgroup of
$G$
. Then M. Hartl identified the subgroup
$G\,\cap \,(1+\,{{\Delta }^{3}}\,(G)\,+\,\Delta (G)\Delta (H))$
of
$G$
. In this note we give an independent proof of the result of Hartl, and we identify two subgroups
$G\,\cap \,(1\,+\,\Delta (H)\Delta (G)\Delta (H)\,+\,\Delta (\left[ H,\,G \right]\Delta (H)),\,G\,\cap \,(1\,+\,{{\Delta }^{2}}\,(G)\Delta (H)\,+\,\Delta (K)\Delta (H))$
of
$G$
for some subgroup
$K$
of
$G$
containing
$[H,G]$
.