We define the stability of a
subgroup under a class of maps, and establish its basic properties. Loosely
speaking, we will say that a normal subgroup, or more generally a normal series
{An} of a group A, is stable under a class of homomorphisms ℋ if whenever
f : A → B lies in ℋ, we have f(a) ∈ Bn if and only if a ∈ An. This translates to
saying that each element of ℋ induces a monomorphism A∕An↪B∕Bn. This
contrasts with the usual theories of localization, wherein one is concerned
with situations where f induces an isomorphism. In the literature, the most
commonly considered classes of maps are those that induce isomorphisms on
(low-dimensional) group homology. The model theorem in this regard is the result of
Stallings that each term of the lower central series is preserved under any
ℤ-homological equivalence of groups. Various other theorems of this nature
have since appeared, involving variations of the lower central series. Dwyer
generalized Stallings’s ℤ results to larger classes of maps, work that was
completed in the other cases by the authors. More recently, the authors
proved analogues of the theorems of Stallings and Dwyer for variations of the
derived series. We interpret all of the theorems above in the framework of
stability.