The Grigorchuk groups, and many other groups of intermediate growth, are residually finite. This need not be the case, as is obvious from Theorem 11.7. Below we will construct, with proofs, other examples, though these groups will still be close to being residually finite. In the other direction we will quote results that show that groups that are not only residually finite, but also residually nilpotent or residually soluble, and have slow subexponential growth, are actually of polynomial growth. We open this chapter with a result that points out that the connection between intermediate growth and residual finiteness may not be accidental. Note that groups of polynomial growth are residually finite, by Theorem 2.24.

Lemma 13.1A finitely generated group contains maximal normal subgroups.

Proof Let G be finitely generated, let S be a finite set which generates G normally, i.e. G = 〈S〉G, and choose S to have minimal possible cardinality. If T contains all elements of S but one, then N ≔ 〈T〉G ≠ G (T may be empty, in which case we take N = 1). By Zorn's Lemma, there exists a normal subgroup K which is maximal with respect to containing N and not containing S. Any normal subgroup containing K properly contains also S, and therefore equals G, thus K is a maximal normal subgroup. QED

Proposition 13.2A finite-by-nilpotent group is nilpotent-by-finite.

Proof Let N ⊲ G, with a finite N and a nilpotent G/N. Let C = CG(N).