  We give a complete characterization of abelian subgroups of GL(n, R) with a
locally dense (resp. dense) orbit in R^n. For finitely generated subgroups,
this characterization is explicit and it is used to show that no abelian
subgroup of GL(n, R) generated by [ (n+1)/2 ] matrices can have a dense orbit
in R^n. ([ ] denotes the integer part).
