Hyper–(Abelian–by–finite) groups with many subgroups of finite depth

The main result of this note is that a finitely generated hyper-(Abelian-byfinite) group G is finite-by-nilpotent if and only if every infinite subset contains two distinct elements x, y such that γn(〈x, x〉) = γn+1(〈x, x〉) for some positive integer n = n(x, y) (respectively, 〈x, x〉 is an extension of a group satisfying the minimal condition on normal subgroups by an Engel group). Groupes hyper-(Abelien-par-fini) ayant beaucoup de sous-groupes de profondeur finie Résumé Le principal résultat de cet article est qu’un groupe G hyper-(Abélien-parfini) de type fini est fini-par-nilpotent si, et seulement si, toute partie infinie de G contient deux éléments distincts x, y tels que γn(〈x, x〉) = γn+1(〈x, x〉) pour un certain entier positif n = n(x, y) (respectivement, 〈x, x〉 est une extension d’un groupe vérifiant la condition minimale sur les sous-groupes normaux par un groupe d’Engel).


Introduction and results
Let X be a class of groups. Denote by (X , ∞) (respectively, (X , ∞) * ) the class of groups G such that for every infinite subset X of G, there exist distinct elements x, y ∈ X such that x, y ∈ X (respectively, x, x y ∈ X ). Note that if X is a subgroup closed class, then (X , ∞) ⊆ (X , ∞) * .
We say that a group G has finite depth if the lower central series of G stabilises after a finite number of steps. Thus if γ n (G) denotes the n th term of the lower central series of G, then G has finite depth if and only if γ n (G) = γ n+1 (G) for some positive integer n. Denote by Ω the class of groups which has finite depth. Moreover, if k is a fixed positive integer, let Ω k denotes the class of groups G such that γ k (G) = γ k+1 (G).
Clearly, any group in the class FN is of finite depth, where F denotes the class of finite groups. From this and the fact that FN is a subgroup closed class, we deduce that finite-by-nilpotent groups belong to (Ω, ∞) * . Here we shall be interested by the converse. In [5], Boukaroura has proved that a finitely generated soluble group in the class (Ω, ∞) is finite-by-nilpotent. We obtain the same result when (Ω, ∞) is replaced by (Ω, ∞) * and soluble by hyper-(Abelian-by-finite). More precisely we shall prove the following result. Note that Theorem 1.1 improves the result of [12] which asserts that a finitely generated soluble-by-finite group whose subgroups generated by two conjugates are of finite depth, is finite-by-nilpotent.
It is clear that an Abelian group G in the class (Ω 1 , ∞) * is finite. For if G is infinite, then it contains an infinite subset X = G\{1}. Therefore there exist two distinct elements x, y ( = 1) in X such that γ 1 ( x, x y ) = γ 2 ( x, x y ) = 1; so x = 1, which is a contradiction. From this it follows that a hyper-(Abelian-by-finite) group G in the class (Ω 1 , ∞) * is hyper-(finite) as (Ω 1 , ∞) * is a subgroup and a quotient closed class. But it is not difficult to see that a hyper-(finite) group is locally finite [17, Part 1, page 36]. So G is locally finite. Now if G is infinite, then it contains an infinite Abelian subgroup A [17,Theorem 3.43]. Since A is in the class (Ω 1 , ∞) * , it is finite; a contradiction and G, therefore, is finite. As consequence of Theorem 1.1, we shall prove other results on the class (Ω k , ∞) * . Corollary 1.2. Let k be a positive integer and let G be a finitely generated hyper-(Abelian-by-finite) group. We have: 3 denotes the class of groups whose 2-generator subgroups are nilpotent of class at most 3.
Let k be a fixed positive integer, denote by M, E k and E respectively the class of groups satisfying the minimal condition on normal subgroups, the class of k-Engel groups and the class of Engel groups. Using Theorem 1.1, we will prove the following results concerning the classes (ME, ∞) * and (ME k , ∞) * Theorem 1.3. Let G be a finitely generated hyper-(Abelian-by-finite) group. Then, G is in the class (ME, ∞) * if, and only if, G is finite-by-nilpotent.
Note that this theorem improves Theorem 3 of [23] (respectively, Corollary 3 of [5]) where it is proved that a finitely generated soluble group in the class (CN , ∞) * (respectively, (X N , ∞)) is finite-by-nilpotent, where C (respectively, X ) denotes the class of Chernikov groups (respectively, the class of groups satisfying the minimal condition on subgroups). Corollary 1.4. Let k be a positive integer and let G be a finitely generated hyper-(Abelian-by-finite) group. We have: (i) If G is in the class (ME k , ∞) * , then there exists a positive integer c = c(k), depending only on k, such that G/Z c (G) is finite.
Note that these results are not true for arbitrary groups. Indeed, Golod [9] showed that for each integer d > 1 and each prime p, there are infinite d-generator groups all of whose (d − 1)-generator subgroups are finite pgroups. Clearly, for d = 3, we obtain a group G which belongs to the class (F, ∞) * . Therefore, G belongs to the classes (Ω, ∞) * , (Ω k , ∞) * , (ME, ∞) * and (ME k , ∞) * , but it is not finite-by-nilpotent.

Proofs of Theorem 1.1 and Corollary 1.2
Let E (∞) the class of groups in which every infinite subset contains two distinct elements x, y such that [x, n y] = 1 for a positive integer n = n(x, y). In [15], it is proved that a finitely generated soluble group in the class E (∞) is finite-by-nilpotent. We will extend this result to finitely generated hyper-(Abelian-by-finite) groups (Proposition 2.5).
Our first lemma is a weaker version of Lemma 11 of [23], but we include a proof to keep our paper reasonably self contained.
Proof. Let G be a finitely generated infinite Abelian-by-finite group in the class (FN , ∞). Hence there is a normal torsion-free Abelian subgroup A of finite index. Let x be a non trivial element in A and let g in G. Then the subset x i g : i a positive integer is infinite, so there are two positive integers m, n such that x m g, x n g is finite-by-nilpotent, hence x r , x n g is finite-by-nilpotent where r = m − n. Thus there are two positive integers c and d such that [x r , c x n g] d = 1. The element x being in A which is Abelian belongs to the torsion-free group A, so [x, c g] = 1. It follows that x is a right Engel element of G. Since G is Abelian-by-finite and finitely generated, it satisfies the maximal condition on subgroups; so the set of right Engel elements of G coincides with its hypercentre which is equal to Z i (G), the (i + 1)-th term of the upper central series of G, for some integer i > 0 [17, Theorem 7.21]. Hence, A ≤ Z i (G); and since A is of finite index in G, G/Z i (G) is finite. Thus, by a result of Baer [10, Theorem 1], G is finite-by-nilpotent.
Proof. Let G be an infinite finitely generated Abelian-by-finite group in E (∞), and let A be an Abelian normal subgroup of finite index in G. It is clear that all infinite subsets of G contains two different elements x, y such that xA = yA; so y = xa for some a in A and x, y = x, a . Thus x, y is a finitely generated metabelian group in the class E (∞). It follows by the result of Longobardi and Maj [15,Theorem 1], that x, y is finite-by-nilpotent. Hence G is in the class (FN , ∞). Now, by Lemma 2.1, G is finite-by-nilpotent; as required.
Proof. Let G be a finitely generated hyper-(Abelian-by-finite) group in the class E (∞). Since E (∞) is a quotient closed class of groups and since finitely generated nilpotent-by-finite groups are finitely presented, we may assume that G is not nilpotent-by-finite but every proper homomorphic image of G is in the class N F. Since G is hyper-(Abelian-by-finite), G contains a non-trivial normal subgroup H such that H is finite or Abelian; so we have G/H is in N F. If H is finite then G is nilpotent-by-finite, a contradiction. Consequently H is Abelian and so G is Abelian-by-(nilpotentby-finite) and therefore it is (Abelian-by-nilpotent)-by-finite. Hence, G is a finite extension of a soluble group; there is therefore a normal soluble subgroup K of G of finite index. Now, K is a finitely generated soluble group in the class E (∞); it follows, by the result of Longobardi and Maj [15,Theorem 1], that K is finite-by-nilpotent. By a result of P. Hall [10, Theorem 2], K is nilpotent-by-finite and so G is nilpotent-by-finite, a contradiction. Now, the Lemma is shown.
Since finitely generated nilpotent-by-finite groups satisfy the maximal condition on subgroups, Lemma 2.3 has the following consequence: Corollary 2.4. Let G be a finitely generated hyper-(Abelian-by-finite) group in the class E (∞). Then G satisfies the maximal condition on subgroups.

Proposition 2.5. A finitely generated hyper-(Abelian-by-finite) group in the class E (∞) is finite-by-nilpotent.
Proof. Let G be a finitely generated hyper-(Abelian-by-finite) group in E (∞). According to Corollary 2.4, G satisfies the maximal condition on subgroups. Now, since E (∞) is a quotient closed class, we may assume that every proper homomorphic image of G is in FN , but G itself is not in FN . Our group G being hyper-(Abelian-by-finite), contains a non-trivial normal subgroup H such that H is finite or Abelian; so by hypothesis G/H is in the class FN . If H is finite, then G is finite-by-nilpotent, a contradiction. Consequently H is Abelian and so G is in the class A (FN ), hence G is in (AF) N . Now, since G satisfies the maximal condition on subgroups, it follows from Lemma 2.2, that G is in (FN ) N , so it is in  F (N N ). Consequently, there is a finite normal subgroup K of G such that G/K is soluble. The group G/K, being a finitely generated soluble group in the class E (∞), is in FN , by the result of Longobardi and Maj [15,Theorem 1]. So G is in the class FN , which is a contradiction and the Proposition is shown.
The remainder of the proof of Theorem 1.1 is adapted from that of Lennox's Theorem [11,Theorem 3] Lemma 2.6. Let G be a finitely generated hyper-(Abelian-by-finite) group in the class (Ω, ∞) * . If G is residually nilpotent, then G is in the class FN .
Proof. Let G be a finitely generated hyper-(Abelian-by-finite) group in the class (Ω, ∞) * and assume that G is residually nilpotent. Let X be an infinite subset of G, there are two distinct elements x and y of X such that x, x y ∈ Ω. It follows that there exists a positive integer k such that γ k ( x, x y ) = γ k+1 ( x, x y ). The group x, x y , being a subgroup of G, is residually nilpotent, so ∩ i∈N γ i ( x, x y ) = 1. Hence γ k ( x, x y ) = ∩ i∈N γ i ( x, x y ) = 1. Since x, x y = [y, x] , x ; γ k ( [y, x] , x ) = 1, thus [y, k x] = 1. We deduce that G is a finitely generated hyper-(Abelian-byfinite) group in the class E (∞). It follows, by Proposition 2.5, that G is in the class FN , as required.

Lemma 2.7. If G is a finitely generated hyper-(Abelian-by-finite) group in the class (Ω, ∞) * , then it is nilpotent-by-finite.
Proof. Let G be a finitely generated hyper-(Abelian-by-finite) group in (Ω, ∞) * . Since finitely generated nilpotent-by-finite groups are finitely presented and (Ω, ∞) * is a quotient closed class of groups, by [17, Lemma 6.17], we may assume that every proper quotient of G is nilpotent-byfinite, but G itself is not nilpotent-by-finite. Since G is hyper-(Abelianby-finite), it contains a non-trivial normal subgroup K such that K is finite or Abelian; so G/K is in N F. In this case, K is Abelian and so G is in the class A(N F) and therefore it is in the class (AN )F. Consequently, G has a normal subgroup N of finite index such that N is Abelian-by-nilpotent. Moreover, N being a subgroup of finite index in a finitely generated group, is itself finitely generated, and so N is a finitely generated Abelian-by-nilpotent group. It follows, by a result of Segal [19,Corollary 1], that N has a residually nilpotent normal subgroup of finite index. Thus, G has a residually nilpotent normal subgroup H, of finite index. Therefore, H is residually nilpotent and it is a finitely generated hyper-(Abelian-by-finite) group in the class (Ω, ∞) * . So, by Lemma 2.6, H is in the class FN , hence H is in the class N F. Thus G is in the class N F, a contradiction which completes the proof. Lemma 2.8. Let G be a finitely generated group in the class (Ω, ∞) * which has a normal nilpotent subgroup N such that G/N is a finite cyclic group. Then G is in the class FN .
Proof. We prove by induction on the order of G/N that G is in the class FN . Let n = |G/N |; if n = 1, then G = N and G is nilpotent. Now suppose that n > 1 and let q be a prime dividing n. Since G/N is cyclic, it has a normal subgroup of index q. Thus G has a normal subgroup H of index q containing N . Since |H/N | < |G/N |, then by the inductive hypothesis, H is in the class FN . Let T be the torsion subgroup of H. Since H is finitely generated, T is finite. So H/T is a finitely generated torsion-free nilpotent group. Therefore, by Gruenberg [18, 5.2.21], H/T is residually a finite p-group for all primes p and hence, in particular, H/T is residually a finite q-group. But H has index q in G from which we get that G/T is residually a finite q-group [20, Exercise 10, page 17]. This means that G/T is residually nilpotent. It follows, by Lemma 2.6, that G/T is in the class FN . So G itself is in FN .
Proof of Theorem 1.1. Let G be a finitely generated hyper-(Abelian-byfinite) group in the class (Ω, ∞) * . Hence, by Lemma 2.7, G is in the class N F. Let K be a normal nilpotent subgroup of G such that G/K is finite. Since K is a finitely generated nilpotent group, it has a normal torsion-free subgroup of finite index [18, 5.4.15 (i)]. Thus, G has a normal torsion-free nilpotent subgroup N of finite index. Let x be a non-trivial element of G. Since N is finitely generated, N, x is a finitely generated hyper-(Abelianby-finite) group in the class (Ω, ∞) * . Furthermore, N, x /N is cyclic. Therefore, by Lemma 2.8, N, x is in the class FN . Consequently, there is a finite normal subgroup H of N, x such that N, x /H is nilpotent. Therefore γ k+1 ( N, x ) ≤ H for some positive integer k; so γ k+1 ( N, x ) is finite. Hence, there is a positive integer m such that [g, k x] m = 1, for all g ∈ N . Since [g, k x] is an element of the torsion-free group N , we get that [g, k x] = 1. Thus, g is a right Engel element of G; so N ⊆ R(G), where R(G) denotes the set of right Engel elements of G. Moreover, since G is a finitely generated nilpotent-by-finite group, it satisfies the maximal condition on subgroups. Therefore, from Baer [17,Theorem 7.21], R(G) coincides with the hypercentre of G which equal to Z n (G) for some positive integer n. Thus N ≤ Z n (G), so Z n (G) is of finite index in G. It follows, by a result of Baer [10, Theorem 1], that G is in the class FN .
Proof of Corollary 1.2. (i) Let G be a finitely generated hyper-(Abelianby-finite) group in the class (Ω k , ∞) * ; from Theorem 1.1, G is in the class FN . Let H be a normal finite subgroup of G such that G/H is nilpotent. It is clear that G/H is in the class (Ω k , ∞) * . LetX be an infinite subset of G/H; there are therefore two distinct elementsx = xH,ȳ = yH (x, y ∈ G) ofX such that x,xȳ ∈ Ω k , so γ k ( x,xȳ ) = γ k+1 ( x,xȳ ). Now, since x,xȳ is nilpotent, there is an integer i such that γ i ( x,xȳ ) = 1; so γ k ( x,xȳ ) = 1. (ii) If G is in the class (Ω 2 , ∞) * , then by Theorem 1.1 G is finite-bynilpotent. Therefore, G has a finite normal subgroup H such that G/H is nilpotent. Since G/H is in the class (Ω 2 , ∞) * , it is in the class E 2 (∞). Hence, by Abdollahi [1,Theorem], (G/H)/Z 2 (G/H) is finite, so γ 3 (G/H) is finite. Since H is finite, γ 3 (G) is finite. It follows, by P. Hall [10, 1.5], that G/Z 2 (G) is finite.
(iii) Now if G is in the class (Ω 3 , ∞) * , then by Theorem 1.1 G has a finite normal subgroup H such that G/H is nilpotent. Since G/H is in the class (Ω 3 , ∞) * , it is in the class E 3 (∞). Hence, by Abdollahi [2, Theorem 1] G/H is in the class FN (2) 3 ; consequently G is in the class FN

Proofs of Theorem 1.3 and Corollary 1.4
We start by showing a weaker version of Theorem 1.3: Proof. Let G be a finitely generated hyper-(Abelian-by-finite) group in the class (MN , ∞) * , and let X be an infinite subset of G. There are therefore two distinct elements x, y of X such that x, x y is in the class MN , so there exists a normal subgroup N of x, x y such that N is in M and x, x y /N is nilpotent. Now, γ i+1 ( x, x y ) ≤ N for some positive integer i, therefore γ i+1 ( x, x y ) ≥ γ i+2 ( x, x y ) ≥ ... is an infinite descending sequence of normal subgroups of N ; however N is in M, therefore there exists a positive integer n ≥ i + 1 such that γ n ( x, x y ) = γ n+1 ( x, x y ). Hence, G is in the class (Ω, ∞) * ; it follows, by Theorem 1.1, that G is finite-by-nilpotent.

Lemma 3.2. A finitely generated hyper-(Abelian-by-finite) group in the class (ME, ∞) * is nilpotent-by-finite.
Proof. Let G be a finitely generated hyper-(Abelian-by-finite) group in the class (ME, ∞) * . Since (ME, ∞) * is a closed quotient class of groups and since finitely generated nilpotent-by-finite groups are finitely presented, we may assume that G is not nilpotent-by-finite, but every proper homomorphic image of G is nilpotent-by-finite. Since G is hyper-(Abelian-byfinite), there exists a non-trivial normal subgroup H of G such that H is finite or Abelian; so we have G/H is nilpotent-by-finite. If H is finite then G is nilpotent-by-finite, a contradiction. Consequently H is Abelian and so G is Abelian-by-(nilpotent-by-finite) and therefore it is (Abelianby-nilpotent)-by-finite. Hence, G is a finite extension of a soluble group. Let K be a normal soluble subgroup of G of finite index. Clearly, K is in (ME, ∞) * , and since all soluble Engel group coincides with its Hirsch-Plotkin radical which is locally nilpotent [17,Theorem 7.34], we deduce that K is in the class (MN , ∞) * ; it follows by Lemma 3.1 that K is finite-by-nilpotent. According to a result of P. Hall [10, Theorem 2], K is nilpotent-by-finite. Thus, G is nilpotent-by-finite, a contradiction. The proof is now complete.
Since finitely generated nilpotent-by-finite groups satisfy the maximal condition on subgroups, Lemma 3.2 has the following consequence: Proof of Theorem 1.3. It is clear that all finite-by-nilpotent groups are in the class (ME, ∞) * . Conversely, let G be a finitely generated hyper-(Abelian-by-finite) group in (ME, ∞) * . According to Corollary 3.3, G satisfies the maximal condition on subgroups. Since Engel groups satisfying the maximal condition on subgroups are nilpotent [18, 12.3.7], we deduce that G is in the class (MN , ∞) * . It follows, by Lemma 3.1, that G is in the class FN ; as required.
Proof of Corollary 1.4. (i) Let G be a finitely generated hyper-(Abelianby-finite) group in the class (ME k , ∞) * ; from Theorem 1.3, G is in the class FN . Let N be a normal finite subgroup of G such that G/N is nilpotent. Since G/N is nilpotent and finitely generated, its torsion subgroup T /N is finite, so T is finite and G/T is a torsion-free nilpotent group. Clearly, the property (ME k , ∞) * is inherited by G/T , and since G/T is torsion-free and soluble, it belongs to (E k , ∞) * [17, Theorem 5.25]. Let X be an infinite subset of G/T ; there are therefore two distinct elements x = xT ,ȳ = yT (x, y ∈ G) ofX such that x,xȳ is a k-Engel group. Since x,xȳ = [ȳ,x] ,x , we have [ȳ, k+1x ] = [[ȳ,x] , kx ] = 1. Hence, G/T is in the class E k+1 (∞). The group G/T , being a finitely generated soluble group in the class E k+1 (∞); it follows by a result of Abdollahi [2,Theorem 3], that there is an integer c = c(k), depending only on k, such that (G/T )/Z c (G/T ) is finite. By a result of Baer [10, Theorem 1], γ c+1 (G/T ) = γ c+1 (G)T /T is finite; and since T is finite, γ c+1 (G) is finite. According to a result of P. Hall [10, 1.5], G/Z c (G) is finite.
(iii) Now if G is in the class (ME 2 , ∞) * , we proceed as in (i) until we obtain that G/T is in the class E 3 (∞). Hence, by Abdollahi [2, Theorem 1] G/T is in the class FN (2) 3 ; consequently G is in the class FN