Generalizing a theorem of P. Hall on finite-by-nilpotent groups

Let $\gamma_i(G)$ and $Z_i(G)$ denote the $i$-th terms of the lower and upper central series of a group $G$, respectively. P. Hall showed that if $\gamma_{i+1}(G)$ is finite then the index $|G:Z_{2i}(G)|$ is finite. We prove that the same result holds under the weaker hypothesis that $|\gamma_{i+1}(G):\gamma_{i+1}(G)\cap Z_i(G)|$ is finite.


Introduction
If G is an arbitrary group, a classical theorem of Schur asserts that if the center Z(G) has finite index in G then derived subgroup G ′ is finite. This was later generalized by Baer (see 14.5.1 in [5]) to any term of the lower central series; namely, if |G : Z i (G)| is finite then γ i+1 (G) is finite. The converse does not hold in general, anyway in [2] P. Hall proved that if γ i+1 (G) is finite then |G : Z 2i (G)| is finite. For the case i = 1 a stronger result is known to hold: actually, if |G ′ : G ′ ∩ Z(G)| is finite then |G : Z 2 (G)| is also finite. This result was obtained independently by the first author and Moretó (see Theorem E of [1]) and by Podoski and Szegedy in [4]. In this paper we show that this last property can be extended to an arbitrary value of i. More precisely, the following is true.
From the proof of the theorem it can be checked that |G : Z 2i (G)| is bounded in terms of |γ i+1 (G) : γ i+1 (G) ∩ Z i (G)|, the ultimate reason being that Lemma 2.1 and Theorem 2.2 below can be stated in a quantitative version. However, we have made no attempt at giving a sharp bound. We will mention here that in the case i = 1, the existence of such a bound was proved by Isaacs in [3] when the group G is finite, and then an explicit bound was given in [4] for an arbitrary group G.
Related to Theorem A, the following two questions arise naturally: (1) To what extent is Theorem A best possible? If the weaker condition that |γ i+1 (G) : γ i+1 (G) ∩ Z i+1 (G)| is finite holds, can we conclude that |G : Z 2i (G)| is finite? (2) In the case that γ i+1 (G) is finite, if G is also finitely generated, then a stronger result holds, namely |G : Z i (G)| is finite. Is this true also under the hypothesis of Theorem A? Does it follow at least that |G : Z j (G)| is finite for some j smaller than 2i? The answer to both these questions is negative. To see this, for arbitrary c, consider a finitely generated nilpotent group G of class c in which the upper and lower central series coincide and such that |G : Z c−1 (G)| is infinite. For example, one can take the semidirect product G = B ⋉ A, where B = b is an infinite cyclic group, A is the free abelian group on free generators a 1 , . . . , a c , and b acts on A by a b i = a i a i+1 for 1 ≤ i ≤ c − 1 and a b c = a c . Now if i ≥ 1 is any fixed integer, we get counterexamples to the first and the second questions by choosing c = 2i + 1 and c = 2i, respectively.
Finally, we observe that combining our result with Baer's theorem it follows that if |γ i+1 (G) : γ i+1 (G) ∩ Z i (G)| is finite then γ 2i+1 (G) is also finite. Actually, one of the key arguments in our proof of Theorem A is the following generalization of this fact, which might be interesting in its own right.
In particular, γ s+t (G) is finite.

The results
The notation we use is standard. Moreover, following the book [5], if A and B are subgroups of a group G and n is a natural number, we define recursively: Throughout the paper, we will repeatedly use the following well-known result (see for instance 5.1.10 in [5]). Another result which will be often used in our proofs is stated for convenience in the following lemma, whose proof is elementary. Most of the times, we will apply it modulo a normal subgroup.
Lemma 2.1. Let H, K be subgroups of a group G. If [H, K] is finite and H is finitely generated, then the centralizer C K (H) has finite index in K.
We will also need the following result of Baer (see for instance 14.5.2 in [5]). The key step in the proof of our main theorem is in the following proposition. Proposition 2.3. Let G be a group and let s ≥ 1 be an integer such that |γ s (G) : As γ s (G)/Z is finite, there exists a finitely generated subgroup U of G such that γ s (G) = γ s (U )Z. By applying P. Hall's theorem to the quotient group G/Z(G), we obtain that |G : Z 2s−1 (G)| is finite. By the theorem of Baer mentioned in the introduction, it follows that γ 2s (G) is finite. Since γ k (U )/γ k+1 (U ) is finitely generated for every k = 1, . . . , 2s − 1, we conclude that all terms of the lower central series of U are finitely generated.
We are going to prove that, for every j = 1, . . . , s, there exists a subgroup We prove the existence of H j by reverse induction on j. For j = s, we take H s = Z. Suppose now that we already have H j+1 of finite index in γ j+1 (G) such that [H j+1 , γ s−j (U )] = 1, and let us see how to construct the subgroup H j . Let K j = C γj (G) (γ s−j (U )Z/Z). Since γ s−j (U )Z/Z is finitely generated and [γ j (G), γ s−j (U )]Z/Z ≤ γ s (G)/Z is finite, it follows from Lemma 2.1 that K j has finite index in γ j (G). Also (1) [ Now we work in the quotient group T j /D j+1 . Since [γ j (G), U ]D j+1 /D j+1 ≤ γ j+1 (G)/D j+1 is finite and U is finitely generated, it follows that the centralizer L j of U D j+1 /D j+1 in γ j (G) has finite index in γ j (G). Observe that (2) [L j , U, γ s−j (U )] ≤ [D j+1 , γ s−j (U )] = 1.
Finally, let H j = K j ∩ L j . Then |γ j (G) : H j | is finite. Moreover, using (1)  Corollary 2.4. Let G be a group such that |γ s (G) : γ s (G)∩Z t (G)| is finite for some s, t. Then |γ s+j (G) : γ s+j (G) ∩ Z t−j (G)| is finite for every j such that 0 ≤ j ≤ t. In particular, γ s+t (G) is finite.
The last part of the proof of Theorem A is inspired from Hall's ideas. The main role will be played by a subgroup C with the two properties that C has finite index in G and [C, s−1 G, C] ≤ Z 2i−s (G) for every s ≥ 1, with the convention that Z j (G) = 1 for j ≤ 0. The following technical lemma will ensure that C has the second property.
Lemma 2.5. Let G be a group and let C j be the centralizer in G of Proof. Observe that C is normal in G and so is [C, k G] for every k. We first prove by induction on k that for all k ≥ 0 and for all ℓ ≥ i + 1.
By definition of C, we have [C, γ ℓ (G)] ≤ Z 2i−ℓ (G) for all ℓ ≥ i + 1 and this settles the case k = 0. Assume now that the statement is true for k. We have Theorem 2.6. Let G be a group such that |γ i+1 (G) : γ i+1 (G) ∩ Z i (G)| is finite. Then |G : Z 2i (G)| is also finite.
Proof. Let C j be the centralizer in G of γ i+j (G)/(γ i+j (G)∩Z i−j (G)) for j = 1, . . . , i. Since |γ i+j (G) : γ i+j (G) ∩ Z i−j+1 (G)| is finite by Corollary 2.4, we can apply Proposition 2.3 to the quotient group G/Z i−j (G), and it follows that |G : C j | is finite. Let C = i j=1 C j , which has also finite index in G. For every s = 1, . . . , i + 1, put K s = [C, s−1 G], which is contained in γ s (G). We prove by reverse induction on s that K s ∩ Z 2i−s+1 (G) has finite index in K s . For s = i + 1 the statement is true, as |K i+1 : K i+1 ∩ Z i (G)| ≤ |γ i+1 (G) : γ i+1 (G) ∩ Z i (G)| is finite by hypothesis. Now assume that Z = K s+1 ∩ Z 2i−s (G) has finite index in K s+1 . As C has finite index in G, we have G = g 1 , . . . , g n , C for some g 1 , . . . , g n ∈ G. Let U = g 1 , . . . , g n and let H s be the centralizer of U Z/Z in K s . Since [K s , U ]Z/Z ≤ K s+1 /Z is finite and U is finitely generated, the subgroup In particular, for s = 1 we obtain that |C : C ∩ Z 2i (G)| is finite. Consequently |G : Z 2i (G)| ≤ |G : C ∩ Z 2i (G)| = |G : C| |C : C ∩ Z 2i (G)| is finite, and we are done.