  Let F be a relatively free algebra of infinite rank. We say that F has the
SMALL INDEX PROPERTY if any subgroup of Gamma=Aut(F) of index at most rank(F)
contains the pointwise stabilizer Gamma_(U) of a subset U of F of cardinality
less than rank(F). We prove that every infinitely generated free
nilpotent/abelian group has the small index property, and discuss a number of
applications.
