  In this paper we study abstract group homomorphisms between the groups of
rational points of linear algebraic groups which are not necessarily reductive.
One of our main goal is to obtain results on homomorphisms from the groups of
rational points of linear algebraic groups defined over certain specific fields
to the groups of rational points of linear algebraic groups over number fields
and non-archimedean local fields of characteristic zero; in this set-up we deal
with the unexplored topic of abstract homomorphisms from the groups of rational
points of anisotropic groups over non-archimedean local fields of
characteristic zero. We also obtain results on abstract homomorphisms from
unipotent and solvable groups, and prove general results on the structures of
abstract homomorphisms using the celebrated result of Borel and Tits in this
area and a well-known theorem due to Tits on the structure of the groups of
rational points of isotropic semisimple groups.
