The Small Index Property For Free Nilpotent Groups

Abstract

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