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On groups whose factor-group modulo the hypercentre has finite section p-rank
Journal of Algebra 440 (2015) 489–503
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On groups whose factor-group modulo the hypercentre has finite section p-rank ✩ M.R. Dixon a , L.A. Kurdachenko b , J. Otal c,∗ a
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA b Department of Algebra, National University of Dnepropetrovsk, 49010 Dnepropetrovsk, Ukraine c Departamento de Matemáticas-IUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain
a r t i c l e
i n f o
Article history: Received 15 January 2015 Available online xxxx Communicated by E.I. Khukhro MSC: 20F14 20F19
a b s t r a c t Several theorems of the following type are obtained: Let G be a locally generalized radical group and let p be a prime. Suppose that G/ζk (G) has finite section p-rank at most r. Then γk+1 (G) has finite section p-rank. Moreover there exists a function τ (r, k) such that rp (γk+1 (G)) ≤ τ (r, k). © 2015 Elsevier Inc. All rights reserved.
Keywords: Finite section p-rank of a group Generalized radical group Hypercentre of a group Locally nilpotent residual of a group
✩ The work of the third author was supported by the Department of I+D of the Government of Aragón (Spain) and FEDER funds from European Union. Some of this work was carried out at the University of Napoli, from which the first two authors received financial support from STAR grant Napoli-24call2013-24. These authors would like to thank the university and its faculty and students for their hospitality during that time. * Corresponding author. E-mail addresses: [email protected] (M.R. Dixon), [email protected] (L.A. Kurdachenko), [email protected] (J. Otal).
http://dx.doi.org/10.1016/j.jalgebra.2015.06.005 0021-8693/© 2015 Elsevier Inc. All rights reserved.
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Introduction It is well-known that if the kth term, ζk (G), of the upper central series of a group G coincides with G, then the (k + 1)th term, γk+1 (G), of the lower central series of G is trivial. R. Baer [1] obtained a nice generalization of this result proving that if the factor group G/ζk (G) is finite, then γk+1 (G) is also finite. This result was extended, using the broad general result [19, Theorem 4.21] and a theorem of Schlette [21], to the class of Chernikov groups, a result stated more explicitly in the paper [18] of J. Otal and J.M. Peña. The class of Chernikov groups is a subclass of the class of groups of finite (special) rank. Here a group G has finite rank r(G) = r if every finitely generated subgroup of G can be generated by r elements and r is the least positive integer with this property. This concept (and the term special rank itself) was introduced by A.I. Maltsev [16]. This result concerning Chernikov groups was itself recently extended to the class of groups of finite rank: in the paper [11] the authors showed that if G/ζk (G) is a locally generalized radical group of finite rank, then γk+1 (G) also has finite rank. We recall that a group G is called generalized radical if it has an ascending series whose factors are either locally nilpotent or locally finite. Accordingly, a group G is locally generalized radical if every finitely generated subgroup of G is generalized radical. We note that the class of locally generalized radical groups is very broad containing, in particular, the classes of locally finite groups and locally soluble groups. Unlike the result of [21], the result of [11] does not hold for all groups since the paper [14] contains an example of a group G whose central factor group is of finite rank but whose derived subgroup is of infinite rank. A further important numerical invariant of a group is its section p-rank, for the prime p, and it is this invariant that concerns us most here. For the prime p we say that a group G has finite section p-rank rp (G) = r if every elementary abelian p-section of G is finite of order at most pr and there is an elementary abelian p-section A/B of G such that |A/B| = pr . If the orders of the elementary abelian p-sections of G are not bounded, then we say that G has infinite section p-rank, and if G has no elementary abelian p-sections, then we will say that rp (G) = 0. Clearly, the concept of section p-rank generalizes the concept of rank. Indeed, if a group G has finite rank r, then G has finite section p-rank for every prime p and rp (G) ≤ r. It is also clear that if G is a torsion-free abelian group of section p-rank at most t, then G has rank at most t. Furthermore, the property of having finite section p-rank is preserved under taking subgroups, factor groups and extensions. Let P be the set of all primes and let σ : P −→ N0 be a function. We shall say that the group G has finite section rank σ if rp (G) = σ(p) for each prime p. Following D.J.S. Robinson [19, Section 9.3], we shall say that a group G has finite abelian section rank if every elementary abelian p-section of G is finite. It is a well-known fact that if a locally generalized radical group has finite abelian section rank, then rp (G) is finite for each prime p. Our first main result is as follows.
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Theorem A. Let G be a locally generalized radical group and let p be a prime. Suppose that G/ζk (G) has finite section p-rank at most r. Then γk+1 (G) has finite section p-rank. Moreover, there exists a function τ (r, k) such that rp (γk+1 (G)) ≤ τ (r, k). The case k = 1 here is [3, Theorem A]. There are a number of interesting consequences of this result which we now list and which mostly follow immediately. Corollary A1. Let G be a locally generalized radical group. Suppose that G/ζk (G) has finite section rank σ, for some function σ : P −→ N0 . Then γk+1 (G) has finite section rank and rp (γk+1 (G)) ≤ τ (σ(p), k), for every prime p. Corollary A2. Let G be a locally generalized radical group. Suppose that G/ζk (G) has finite abelian section rank. Then γk+1 (G) has finite abelian section rank. Corollary A3. Let G be a locally generalized radical group. Suppose that G/ζk (G) has finite rank r. Then γk+1 (G) has finite rank, and there is a function τ1 (r, k) such that r(γk+1 (G)) ≤ τ1 (r, k). Corollary A4. Let G be a finite group. Suppose that G/ζk (G) has rank r. Then there exists a function τ1 (r, k) such that r(γk+1 (G)) ≤ τ1 (r, k). Suppose that X is a class of groups. For the group G we let GX = ∩{H G|G/H ∈ X}, / X, but in many important classes of groups the X-residual of G. In general G/GX ∈ membership does hold. We shall let GN denote the nilpotent residual of a group G and let GLN denote the locally nilpotent residual of G. In the paper [6], Baer’s theorem was generalized in a different direction. There the authors proved that if G is a group whose hypercentre has finite index, then the locally nilpotent residual, GLN , of G is finite and G/GLN is hypercentral. In [13] a quantitative version of this result was given. Here we provide a substantial generalization of Baer’s theorem. We shall prove: Theorem B. Let G be a group and let p be a prime. Suppose that the upper hypercentre of G contains a G-invariant subgroup Z such that G/Z is locally finite and has finite section p-rank r. Then the locally nilpotent residual, GLN , of G is locally finite, Π(GLN ) ⊆ Π(G/Z), and there is a function τ2 (r) such that rp (GLN ) ≤ τ2 (r). Moreover, G/GLN is locally nilpotent. The following consequences can be deduced. We observe that the groups in the question are generalized radical groups.
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Corollary B1. Let G be a group. Suppose that the hypercentre of G contains a G-invariant subgroup Z such that G/Z is locally finite and has finite section rank σ, for some function σ : P −→ N0 . Then GLN is locally finite, Π(GLN ) ⊆ Π(G/Z) and rp (GLN ) ≤ τ2 (σ(p)), for all primes p. Moreover, G/GLN is hypercentral. This particular corollary can be rephrased as follows. Corollary B2. Let G be a group. Suppose that the hypercentre of G contains a G-invariant subgroup Z such that G/Z is locally finite and has finite abelian section rank. Then GLN is locally finite and has finite abelian section rank. Moreover, G/GLN is hypercentral. Corollary B3. Let G be a group. Suppose that the hypercentre of G contains a G-invariant subgroup Z such that G/Z is locally finite and has finite rank r. Then GLN is locally finite, Π(GLN ) ⊆ Π(G/Z), and there exists a function τ3 such that r(GLN ) ≤ τ3 (r). Moreover, G/GLN is hypercentral. Our next result has also appeared in [10] for periodic groups. Corollary B4. Let G be a group. Suppose that the hypercentre of G contains a G-invariant subgroup Z such that G/Z is a Chernikov group. Then GLN is also Chernikov. Furthermore, G/GLN is hypercentral. The condition that G/Z is locally finite in Theorem B and its corollaries is essential. In Section 4 we give an example to show that the result breaks down completely in the absence of this hypothesis. Finally, A. Mann [17] proved that if G/ζ(G) is locally finite of exponent e, then γ2 (G) is also locally finite of finite exponent m(e), for some function m. This result was generalized in the paper [12], where it is proved that if G/ζk (G) is locally finite of exponent e, then γk+1 (G) is also locally finite of finite exponent. Our final main result gives yet another generalization of Mann’s theorem. Theorem C. Let G be a group. Suppose that the hypercentre of G contains a G-invariant subgroup Z such that G/Z is locally finite and has exponent e. Then the locally nilpotent residual, GLN , of G is locally finite of exponent at most β2 (e), for some function β2 depending upon e only. Our notation is standard. We let ζ∞ (G) denote the hypercentre of the group G. 1. Proof of Theorem A In this section we shall provide the proof of Theorem A and first briefly discuss the class of locally generalized radical groups, which has been studied in several recent papers. A result that is fundamental in much of what follows concerns the structure of
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such groups with finite section p-rank at most r. For convenience we include this result here and note that the proof occurs in both [3] and [5]. Theorem 1.1. Let G be a locally generalized radical group and let p be a prime. If every finitely generated subgroup of G has finite section p-rank at most r, then G has normal subgroups 1 ≤ T ≤ L ≤ K ≤ S ≤ G such that (i) (ii) (iii) (iv)
T is locally finite and G/T is soluble-by-finite of finite rank; L/T is a torsion-free nilpotent group of rank at most r; K/L is a finitely generated torsion-free abelian group of rank at most r; G/K is finite of order at most θ1 (r) and S/T is the soluble radical of G/T of derived length at most f1 (r).
Furthermore, the Sylow p-subgroups of T are Černikov of finite rank at most f2 (r), for some function f2 : N −→ N. It follows easily from this theorem that the group G/T has rank at most θ2 (r) = 2r +log2 (θ1 (r)). Of course we have no control over the p -subgroups of T in this theorem. It was also shown in [5, Theorem D] that if G is a locally generalized radical group and if there is an integer r such that r(A) ≤ r for all abelian subgroups A, then G has finite rank at most f (r) + r(5r + 1)/2 + 1, for some function f of r only. It is easy to extract the following result from the proof of [5, Theorem D]. Lemma 1.2. Let G be a locally finite group and suppose that the abelian subgroups of G have rank at most r. Then G has rank at most r(5r + 1)/2 + 1. With these preliminaries taken care of we now show how to control the structure of certain factor groups. Next we quote the following result whose proof appears in [3, Lemma 3.10]. Lemma 1.3. Let G be a group, let p be a prime and let A be a normal abelian p-subgroup of G. Suppose that G satisfies the following conditions: (i) G/CG (A) is a periodic abelian p -group; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank r. Then G/CG (A) is finite of rank at most r. We shall prove certain variations of this type of result, the first one being as follows. Lemma 1.4. Let G be a group, let p be a prime and let A be a normal abelian p-subgroup of G. Suppose that G satisfies the following conditions:
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(i) G/CG (A) is a locally finite group of finite section p-rank t; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank r. Then G/CG (A) is Chernikov of rank at most t + r(5r + 1)/2 + 1, and if t = 0 then G/CG (A) is finite. Proof. Let V = G/CG (A). If Q is a p -subgroup of V then, by Lemma 1.3, the abelian subgroups of Q are finite of rank at most r. By a theorem of Hall and Kulatilaka [7], every infinite locally finite group contains an infinite abelian subgroup, so Q must be finite. In particular, if t = 0, then V must be finite. If T is a Sylow p-subgroup of V , then T has finite section p-rank at most t and hence T has rank at most t by [3, Corollary 2.3]. Therefore, T is a Chernikov group by [19, Corollary 2]. It follows that V is a locally finite group with Chernikov Sylow q-subgroups for all primes q and a theorem of Belyaev (see [4, Theorem 3.5.15]) implies that V is almost locally soluble. Hence V /Op (V ) is a Chernikov group (see [4, Theorem 2.5.12]). Since Op (V ) is finite, by our work above, V is Chernikov. Finally, let F be an arbitrary finite subgroup of V . If S is a Sylow q-subgroup of F , for some prime q = p, then S is finite and has rank at most ρ(r) = r(5r + 1)/2, by [20, Corollary 5], so S has at most ρ(r) generators. If S is a Sylow p-subgroup of F , then S has rank at most t, so S has at most t-generators. By [15, Theorem 1], it follows that F has at most max(t, ρ(r)) + 1 generators. We deduce that G/CG (A) has rank at most t + ρ(r) + 1, as required. 2 Our next result shows that the property of having bounded section p-rank is really a local property. Lemma 1.5. Let G be a group, let L be a local system for G and let p be a prime. If there exists a positive integer t such that the section p-rank of every subgroup H ∈ L is at most t, then G has section p-rank at most t. Proof. Let U, V be subgroups of G such that U V and V /U is a finite elementary abelian p-group. Then V /U = a1 U × . . . × an U , for certain elements a1 , a2 , . . . , an of V such that api ∈ U for each i, where 1 ≤ i ≤ n. Since L is a local system, there is subgroup Ki ∈ L such that ai ∈ Ki , for 1 ≤ i ≤ n and, for the same reason, there is a subgroup Y ∈ L such that Ki ≤ Y , for 1 ≤ i ≤ n. Then A = a1 , . . . , an ≤ Y . Now V = AU so V /U ∼ = A/A ∩ U . Since Y ∈ L, it has section p-rank at most t and, since A ≤ Y , it follows that the elementary abelian p-sections of A have rank at most t. Hence n ≤ t and it follows that rp (G) ≤ t. 2
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We now start to transfer hypotheses concerning factor groups to conclusions about commutators. Lemma 1.6. Let G be a group, let p be a prime and let A be a normal abelian p-subgroup of G. Suppose that G satisfies the following conditions: (i) G/CG (A) = x1 CG (A), . . . , xm CG (A) , where x1 , . . . , xm ∈ G; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank t. Then [A, G] has finite section p-rank at most tm. Proof. Let g be an arbitrary element of G and let ζg : A −→ [A, g] be the endomorphism defined by ζg (a) = [a, g]. Since B ≤ ζ(G) ∩ A ≤ CA (g) = ker(ζg ) it follows that A/CA (g) has finite section p-rank at most t. However A/CA (g) ∼ = [A, g], so rp ([A, g]) ≤ t also. By [11, Lemma 1.1] we have [A, G] = [A, x1 ] . . . [A, xm ] and it then follows easily that rp ([A, G]) ≤ tm, as required. 2 This yields the following result, which is analogous to [11, Lemma 1.2]. Lemma 1.7. Let G be a group, let p be a prime and let A be a normal abelian p-subgroup of G. Suppose that G satisfies the following conditions: (i) G/CG (A) is a group of finite rank t; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank r. Then [A, G] has finite section p-rank at most tr. Proof. Let L be the local family consisting of all the finitely generated subgroups of G/CG (A). If F/CG (A) ∈ L then F/CG (A) = x1 CG (A), . . . , xm CG (A) , for certain elements x1 , . . . , xm of G such that m ≤ t. Lemma 1.6 implies that [A, F ] has finite section p-rank at most rm ≤ rt. Of course {[A, F ]|F/CG (A) ∈ L} is a local system for [A, G] and Lemma 1.5 implies that rp ([A, G]) ≤ tr. 2 We require further information concerning normal abelian subgroups and, for our next result, we recall that if T is a periodic subgroup of GLn (Q), then T is finite and there is a function ρ, of n only, such that |T | ≤ ρ(n) (see [22, Theorem 9.33], for example). From this it follows that a periodic group of automorphisms of a torsion-free abelian group of finite Z-rank n is finite of order at most ρ(n). Lemma 1.8. Let G be a group, let p be a prime and let A be a normal torsion-free abelian subgroup of G. Suppose that
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(i) G/CG (A) is periodic; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank r. Then G/CG (A) is finite and has order at most ρ(r). Proof. Let T /B be the torsion subgroup of A/B and let a ∈ T , so there is a positive integer k such that ak ∈ B. If g ∈ G then (ag )k = (ak )g = ak . Since A is torsion-free, it follows that ag = a and hence T ≤ ζ(G). Now, A/T is torsion-free and hence has rank at most r, by our hypotheses. Let U = CG (A/T ) and note that, by our remark above, G/U has order at most ρ(r). If a ∈ A, u ∈ U , then au = ac, for some element c ∈ T . Since G/CG (A) is periodic, there is a positive integer m such that um ∈ CG (A), so a = u−m aum = acm . It follows that cm = 1 and, since A is torsion-free, we deduce that c = 1. Hence U = CG (A/T ) = CG (A) and the result now follows. 2 In our next result we require structural information about locally generalized radical groups, as elucidated in Theorem 1.1. Lemma 1.9. Let G be a locally generalized radical group, let p be a prime and let A be an abelian normal subgroup of G. Suppose that (i) G/CG (A) has finite section p-rank t; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank r; (iii) The torsion subgroup of A is a p-group. Then [A, G] has finite rank and there is a function θ4 such that r([A, G]) ≤ θ4 (t, r). Proof. Let Q/B be the Sylow p -subgroup of A/B and let a ∈ Q. Then there is a p -number k such that ak ∈ B. If g ∈ G, then (ag )k = (ak )g = ak and, since A is abelian, we have (a−1 ag )k = 1. However, the hypothesis on A implies that ag = a so we deduce that Q ≤ ζ(G). The choice of Q implies that A/Q contains no p -elements. Since A/Q has p-rank at most r, [3, Lemma 2.1] shows that A/Q has finite rank at most r. Let T be the torsion subgroup of A and let R/CG (A) be the largest normal torsion subgroup of G/CG (A). By Lemma 1.4, R/CR (T ) is a Chernikov group of rank at most t + r(5r + 1)/2 + 1. Furthermore, Lemma 1.8 shows that R/CR (A/T ) is finite of order at most ρ(r). By Remak’s theorem, we have the embedding R/(CR (T ) ∩ CR (A/T )) −→ R/CR (T ) × R/CR (A/T ), which shows that R/(CR (T ) ∩ CR (A/T )) is Chernikov of rank at most t + r(5r + 1)/2 + 1 + log2 (ρ(r)). Now let a ∈ A and x ∈ CR (T ) ∩ CR (A/T ). Then ax = ac for some c ∈ T . Since T is a p-group, there is a p-number m such that cm = 1. Then x−m axm = acm = a so xm ∈ CG (A). Hence CR (T ) ∩ CR (A/T )/CG (A) is a p-group. By our hypotheses and
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[3, Corollary 2.3] we deduce that CR (T ) ∩ CR (A/T )/CG (A) is Chernikov of rank at most t. Hence R/CG (A) is Chernikov of rank at most 2t + r(5r + 1)/2 + 1 + log2 (ρ(r)). By Theorem 1.1, G/R has rank at most θ2 (t) and hence G/CG (A) has rank at most θ2 (t) + 2t + r(5r + 1)/2 + 1 + log2 (ρ(r)) = θ3 (t, r). Applying [11, Lemma 1.2], we deduce that [A, G] has finite rank at most rθ3 (t, r) = θ4 (t, r), as required. 2 Proposition 1.10. Let G be a locally generalized radical group, let p be a prime and let A be a normal abelian subgroup of G. Suppose that G satisfies the following conditions: (i) G/CG (A) has finite section p-rank t; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank r. Then [A, G] has finite section p-rank at most θ4 (t, r). Proof. Let Q be the Sylow p -subgroup of A. By Lemma 1.9, [A/Q, G/Q] = V /Q has finite rank at most θ4 (t, r) and hence V also has finite section p-rank at most θ4 (t, r). However, [A, G] ≤ V so the result follows. 2 Proof of Theorem A. Let 1 = Z0 ≤ Z1 ≤ . . . ≤ Zk−1 ≤ Zk = Z be the upper central series of G. We use induction on k. If k = 1, then the central factor group G/Z1 has finite section p-rank r and [3, Theorem A] implies that γ2 (G) has finite section p-rank and that there is a function λ2 such that rp (γ2 (G)) ≤ λ2 (r). Assume now that k ≥ 1 and apply our induction hypothesis to G/Z1 , so that there is a function τ such that rp (γk (G/Z1 )) ≤ τ (r, k − 1). Set K/Z1 = γk (G/Z1 ), let L = γk (G) and note that K = LZ1 . Then K = [LZ1 , LZ1 ] = L , since Z1 = Z(G). Also, [K, G] = [L, G] = γk+1 (G). Theorem A of [3] shows that L has finite section p-rank and rp (L ) ≤ λ2 (τ (r, k − 1)). Well-known properties of the upper and lower central series show that [L, Z] = 1, so G/CG (L) is an image of G/Z and hence rp (G/CG (L)) ≤ r. We wish to apply Proposition 1.10 to G/L . To this end we note that L/L ∩Z1 L /L ≤ L/L ∩ ζ(G/L ) and that (L/L )/(L/L ∩ Z1 L /L ) ∼ = (L/L )(Z1 L /L )/(Z1 L /L ) ∼ = LZ1 /L Z1 ∼ = L/(L ∩ Z1 L ), an image of L/L ∩ Z1 . Certainly, rp (L/L ∩ Z1 ) = rp (LZ1 /Z1 ) ≤ rp (K/Z1 ) ≤ τ (r, k − 1). Also, (G/L )/CG/L (L/L ) is an image of G/CG (L), so rp ((G/L )/CG/L (L/L )) ≤ r.
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Then, by Proposition 1.10, [L/L , G/L ] = [L, G]/L = γk+1 (G)/L has finite section p-rank at most θ4 (r, τ (r, k − 1)). It follows that γk+1 (G) has finite section p-rank and that rp (γk+1 (G)) ≤ λ2 (τ (r, k − 1)) + θ4 (r, τ (r, k − 1)) = τ (r, k).
2
Proposition 1.11. Let G be a locally generalized radical group. If there exists a positive integer r such that rp (G) ≤ r, for each prime p, then G has finite rank at most 4r + 2. Proof. By Theorem 1.1, G has a series 1 ≤ T ≤ L ≤ K ≤ S ≤ G. The group L/T has rank at most r, by [24] and K/L is also of rank at most r. Hence K/T has rank at most 2r. We note that T is locally finite. Let F be a finite subgroup of T , let q ∈ Π(F ) and let S be a Sylow q-subgroup of F . Then [3, Lemma 2.2] shows that S has rank at most r, so that S has at most r generators. Since this is true for each prime q, it follows from [15, Theorem 1] that F has at most r + 1 generators. Hence T has rank at most r + 1. The same argument applies in the finite factor group G/K, so that it has rank at most r + 1 too. Hence G has rank at most 2(r + 1) + 2r = 4r + 2, as required. 2 Proof of Corollary A3. Suppose that G/ζk (G) has finite rank r. As we saw above, this means that rp (G/ζk (G)) ≤ r, for each prime p. Theorem A then implies that rp (γk+1 (G)) ≤ τ (r, k), for each prime p. By Proposition 1.11, γk+1 (G) has rank at most 4τ (r, k) + 2 = τ1 (r, k). 2 2. Proof of Theorem B Let G be a group whose centre contains a subgroup Z such that G/Z is finite of order t. Then |G/ζ(G)| ≤ t and hence, by a result of J. Wiegold [23], we have |G | ≤ w(t) = tm , where m = (1/2)(logp t − 1) and p is the least prime dividing t. If t1 = |G/ζ(G)| then t1 is a divisor of t and it follows that Π(G/ζ(G)) ⊆ Π(G/Z). It follows from [2, Zusatz] that the orders of the elements of G divide t21 and hence divide t2 . Hence Π(G ) ⊆ Π(G/Z). If α is a real number then we let ι(α) denote the least integer not less than α. Lemma 2.1. Let p be a prime, let G be a finite group, let Z be a G-invariant subgroup of the hypercentre of G and let GN be the nilpotent residual of G. Then Π(GN ) ⊆ Π(G/Z) and there exist functions τ2 , τ3 such that rp (GN ) ≤ τ2 (rp (G/Z)) and |GN | ≤ τ3 (|G/Z|). Proof. Let |G/Z| = t, r = rp (G/Z) and π = Π(G/Z). There is a series of G-invariant subgroups 1 = Z0 ≤ Z1 ≤ . . . ≤ Zn ≤ Zn+1 = Z
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whose factors are G-central. Let C = CG (Z). By a theorem of L.A. Kaluzhnin [9], G/C is nilpotent and clearly Z ≤ CG (C), so that |G/CG (C)| ≤ t. We have C/Z ∩ C ∼ = CZ/Z, so C/C ∩Z has order dividing t, rp (C/Z ∩C) ≤ r and Π(C/Z ∩C) ⊆ π. Since C ∩Z ≤ ζ(C), the remark preceding the lemma shows that |C | ≤ w(t) and Π(C ) ⊆ π. Applying [3, Proposition 3.3] to C shows that there exists a function λ such that rp (C ) ≤ λ(r) = r(3r + 1)/2 + r2 ι(log2 r). Since G/C is nilpotent, so is (G/C )/CG/C (C/C ) and [11, Corollary 2.5] shows that C/C has a Z-decomposition, say ∞ ∞ C/C = ζZG (C/C ) ⊕ EZG (C/C ). ∞ Clearly (C ∩ Z)C /C ≤ ζZG (C/C ) from which it follows that the order of R/C = ∞ EZG (C/C ) is a divisor of t, rp (R/C ) ≤ r and Π(R/C ) ⊆ π. Furthermore, the factor C/R ∼ = (C/C )/(R/C ) is G-hypercentral and hence the hypercentre of G/R contains C/R. Since G/C is nilpotent, G/R is therefore nilpotent and hence GN ≤ R. Finally,
|R| = |C | · |R/C | ≤ w(t)t = tm+1 = τ3 (t), rp (R) ≤ r + λ(r) = τ2 (r) and Π(R) ⊆ π. The result follows. 2 If G is a group, then we let Tor(G) denote the maximal normal periodic subgroup of G and note that, when G is locally nilpotent, Tor(G) is a characteristic subgroup of G such that G/Tor(G) is torsion-free. Before proving the next result we make the following observation. Suppose that G is a finitely generated group and suppose that Z is a G-invariant subgroup of the hypercentre of G such that G/Z is finite. Then ζ∞ (G) is also finitely generated, and locally nilpotent. Hence ζ∞ (G) has the maximum condition and it follows that there is a positive integer n such that ζ∞ (G) = ζn (G). Baer’s Theorem [1] implies that γn+1 (G) is finite. Hence GN is finite and G/GN is nilpotent. Lemma 2.2. Let p be a prime, let G be a finitely generated group, and let Z be a G-invariant subgroup of the hypercentre of G. Suppose that G/Z is finite. Then GN is finite and Π(GN ) ⊆ Π(G/Z). Also, rp (GN ) ≤ τ2 (rp (G/Z)) and |GN | ≤ τ3 (|G/Z|). Furthermore, G/GN is nilpotent. Proof. Let |G/Z| = t, r = rp (G/Z) and π = Π(G/Z). As we remarked above, the nilpotent residual GN is finite and G/GN is nilpotent. Since G/GN is finitely generated, its torsion subgroup K/GN is finite and hence G/K is a finitely generated torsion-free nilpotent group. Since Z is a finitely generated nilpotent subgroup, its torsion subgroup T is finite. By [8], Z contains a normal torsion-free subgroup U such that the orders of the elements
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of Z/U are the divisors of some positive integer k. Let V = Z k ≤ U , so that V is also torsion-free. Clearly, V is G-invariant and G/V is periodic. However, G/V is finitely generated and nilpotent-by-finite, so G/V must be finite. The hypercentre of G/V contains the G-invariant subgroup ZV /V . Furthermore, |(G/V )/(ZV /V )| ≤ t, rp ((G/V )/(ZV /V )) ≤ r and Π((G/V )/(ZV /V )) ⊆ π. Let D/V be the nilpotent residual of the finite group G/V . By Lemma 2.1, |D/V | ≤ τ3 (t), rp (D/V ) ≤ τ2 (r) and Π(D/V ) ⊆ π. Since G/K and G/D are nilpotent, we have that G/(K ∩ D) is also nilpotent and hence GN ≤ K ∩ D. Furthermore, V is torsion-free and K is finite, so V ∩ K = 1. Then K ∩D ∼ = (K ∩ D)V /V ≤ D/V. It follows that |K ∩ D| ≤ τ3 (t), rp (K ∩ D) ≤ τ2 (r) and Π(K ∩ D) ⊆ π. Since GN ≤ K ∩ D this completes the proof. 2 Proof of Theorem B. Let L be the family of all finitely generated subgroups of G. If U ∈ L, then the hypercentre of U contains the U -invariant subgroup U ∩ Z = ZU and U/ZU is a finitely generated, locally finite group. Hence U/ZU is finite. Also, since U/ZU ∼ = U Z/Z ≤ G/Z, we have rp (U/ZU ) ≤ r and Π(U/ZU ) ⊆ π. Lemma 2.2 shows that U N is finite, rp (U N ) ≤ τ2 (r) and Π(U N ) ⊆ π. Furthermore, U/U N is nilpotent. Now let V be another subgroup in L. Then there exists W ∈ L such that U, V ≤ W . Of course, W N ∩ U U and U/(W N ∩ U ) is nilpotent. Hence U N ≤ W N and, similarly, V N ≤ W N . It follows that R = U ∈L U N is a locally finite subgroup of G such that Π(R) ⊆ π. By Lemma 1.5, rp (R) ≤ τ2 (r). Also, R is a normal subgroup of G, since U N is a normal subgroup of U for each U ∈ L. Let F/R be a finitely generated subgroup of G/R. Then there is a finitely generated subgroup V of G such that F = V R. Of course, V N ≤ R ∩ V and hence V /R ∩ V ∼ = V R/R = F/R is nilpotent. Consequently, G/R is locally nilpotent and it follows that GLN ≤ R. Clearly, Π(GLN ) ⊆ π and rp (GLN ) ≤ τ2 (r). Next, we let Y be a normal subgroup of G such that G/Y is locally nilpotent. If U ∈ L, then U Y /Y is nilpotent and hence U N ≤ U ∩ Y . In particular U N ≤ Y , and we have R = U ∈L U N ≤ Y . Since this is true for all such normal subgroups Y , we have R ≤ GLN and hence R = GLN . Hence G/GLN is locally nilpotent. This completes the proof. 2 Proof of Corollaries B1–B4. For Corollary B1 we suppose that G has finite section rank σ. By Theorem B, G/GLN is locally nilpotent. Let K/GLN be the hypercentre
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of G/GLN . Then G/K is locally nilpotent. However, ZGLN /GLN ≤ K/GLN , so G/K is also locally finite and has finite section rank. The p-component of G/K is Chernikov and hence hypercentral, using [3, Corollary 2.3] and well-known facts. It follows that G/K is hypercentral, so that G = K. Hence Corollaries B1 and B2 follow. For Corollary B3, it follows from Corollary B1 that rp (GLN ) ≤ τ2 (r), for all primes p. It follows from Proposition 1.11 that GLN is of rank at most τ3 (r) = 4τ2 (r) + 2. For Corollary B4, we note that GLN has finite rank, by Corollary B3, and that there are only finitely many primes in Π(GLN ). Hence G is Chernikov. 2 3. Proof of Theorem C Let L be the family of all finitely generated subgroups of G. If U ∈ L, then the hypercentre of U contains the U -invariant subgroup U ∩ Z = ZU . Furthermore, U/ZU is finitely generated and locally finite. Hence U/ZU is finite and, since U/ZU ∼ = U Z/Z ≤ G/Z, the exponent of U/ZU is at most e. By our remark before the proof of Lemma 2.2, the nilpotent residual U N of U is finite and, by [12, Theorem B], U N has finite exponent β2 (e), for some function β2 of e only. Now let V be another subgroup in L. Then there exists W ∈ L such that U, V ≤ W . Of course, W N ∩ U U and U/(W N ∩ U ) is nilpotent. Hence U N ≤ W N and similarly V N ≤ W N . Hence R = U ∈L U N is a locally finite subgroup of G. Since every subgroup of U N has finite exponent at most β2 (e), R also has finite exponent at most β2 (e). As in the proof of Theorem B, R contains GLN . This completes the proof. 4. An example Finally, we give the example promised in the Introduction. Let G = a d be an infinite dihedral group, where a is an element of infinite order, d has order 2 and ad = a−1 . Let A0 = a and, for each n ≥ 0, let An+1 = A2n . Each of the subgroups An is clearly normal in G and G/An is a finite dihedral group of order 2n+1 . In the group Dn = G/An let an = aAn and vn = dAn . Let b be an element of order 2 and let Kn = b Dn , the wreath product of b by Dn . Let Bn denote the base group of this wreath product, so that Kn = Bn Dn . Let B = Drn∈N Bn denote the direct product of the groups Bn , let D = Crn∈N Dn denote the Cartesian product of the groups Dn and define an action of D on B as follows: If b = (bn ) ∈ Drn∈N Bn and y = (dn ) ∈ Crn∈N Dn let by = (bdnn ). Let H = B D be the corresponding semidirect product defined by this action. The mapping f : G −→ D defined by f (x) = (xAn )n∈N is clearly a homomorphism and, since ∩n∈N An = 1, Remak’s theorem implies that this homomorphism is an embedding of G into D. Thus we may form the subgroup of H defined by L = B f (G).
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A short calculation (which we omit) shows that B = ζ∞ (L) and that the length of the upper central series of L is the first infinite ordinal, ω. Hence L/ζ∞ (L) ∼ = f (G) ∼ =G is infinite dihedral, so is of finite rank. We suppose, for a contradiction, that L contains a normal subgroup P such that P has finite rank and L/P is locally nilpotent. Since B is an elementary abelian 2-group and L/B is polycyclic, it follows that P is also polycyclic. If P is periodic, then P is finite, so f (G) ∩ P is a finite normal subgroup of the infinite dihedral group f (G). This implies that f (G) ∩ P = 1 and hence f (G)P/P ∼ = f (G) is locally nilpotent, which is a contradiction. Hence P is non-periodic and, since L has torsion-free rank 1, it follows that P has torsion-free rank 1. Since P is polycyclic, it contains an infinite cyclic normal subgroup of finite index and hence a characteristic such subgroup u . Since L = B ·f (G) and f (G) is dihedral, we have u = xf (as ) for some element x ∈ B and some integer s. Since x ∈ ζk (L), for some natural number k, it is easy to see that, for a suitable positive power l, we have ul = f (asl ). However, ul has infinitely many conjugates in L whereas, by construction, ul is a normal subgroup of L. This gives us the required contradiction and it follows that L has no normal subgroup of finite rank with locally nilpotent factor group. References [1] R. Baer, Representations of groups as quotient groups. I, Trans. Amer. Math. Soc. 58 (1945) 295–347. [2] R. Baer, Endlichkeitskriterien für Kommutatorgruppen, Math. Ann. 124 (1952) 161–177. [3] A. Ballester-Bolinches, S. Camp-Mora, L.A. Kurdachenko, J. Otal, Extension of a Schur theorem to groups with a central factor with a bounded section rank, J. Algebra 393 (2013) 1–15. [4] M.R. Dixon, Sylow Theory, Formations and Fitting Classes in Locally Finite Groups, Ser. Algebra, vol. 2, World Scientific, Singapore, 1994. [5] M.R. Dixon, L.A. Kurdachenko, N.V. Polyakov, Locally generalized radical groups satisfying certain rank conditions, Ric. Mat. 56 (2007) 43–59. [6] M. De Falco, F. de Giovanni, C. Musella, Y.P. Sysak, On the upper central series of infinite groups, Proc. Amer. Math. Soc. 139 (2) (2011) 385–389. [7] P. Hall, C.R. Kulatilaka, A property of locally finite groups, J. London Math. Soc. 39 (1964) 235–239. [8] H. Heineken, L.A. Kurdachenko, Groups with subnormality for all subgroups that are not finitely generated, Ann. Mat. Pura Appl. 169 (1995) 203–232. [9] L.A. Kaluzhnin, Über gewisse Beziehungen zwischen einer Gruppe und ihren Automorphismen, in: Bericht über die Mathematiker-Tagung in Berlin, Januar, 1953, Deutscher Verlag der Wissenschaften, Berlin, 1953, pp. 164–172. [10] L.A. Kurdachenko, J. Otal, Groups with Chernikov factor-group by hypercentral, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM (2015), http://dx.doi.org/10.1007/ s13398-014-0201-7. [11] L.A. Kurdachenko, J. Otal, The rank of the factor-group modulo the hypercenter and the rank of the some hypocenter of a group, Cent. Eur. J. Math. 11 (10) (2013) 1732–1741. [12] L.A. Kurdachenko, J. Otal, A.A. Pypka, Relationships between the factors of the upper and lower central series of a group, Bull. Malaysian Math. Sci. Soc. (2015), in press. [13] L.A. Kurdachenko, J. Otal, I.Ya. Subbotin, On a generalization of Baer theorem, Proc. Amer. Math. Soc. 141 (8) (2013) 2597–2602. [14] L.A. Kurdachenko, P. Shumyatsky, The ranks of central factor and commutator groups, Math. Proc. Cambridge Philos. Soc. 154 (1) (2013) 63–69. [15] A. Lucchini, A bound on the number of generators of a finite group, Arch. Math. (Basel) 53 (4) (1989) 313–317. [16] A.I. Maltsev, On groups of finite rank, Mat. Sb. 22 (1948) 351–352.
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[17] A. Mann, The exponents of central factor and commutator groups, J. Group Theory 10 (4) (2007) 435–436. [18] J. Otal, J.M. Peña, Nilpotent-by-Černikov CC-groups, J. Austral. Math. Soc. Ser. A 53 (1) (1992) 120–130. [19] D.J.S. Robinson, Finiteness Conditions and Generalized Soluble Groups, vols. 1 and 2, Ergeb. Math. Grenzgeb., vols. 62, 63, Springer-Verlag, Berlin, Heidelberg, New York, 1972. [20] D.J.S. Robinson, A new treatment of soluble groups with a finiteness condition on their abelian subgroups, Bull. Lond. Math. Soc. 8 (1976) 113–129. [21] A. Schlette, Artinian, almost abelian groups and their groups of automorphisms, Pacific J. Math. 29 (1969) 403–425. [22] B.A.F. Wehrfritz, Infinite Linear Groups, Ergeb. Math. Grenzgeb., vol. 76, Springer-Verlag, New York, Heidelberg, Berlin, 1973. [23] J. Wiegold, Multiplicators and groups with finite central factor-groups, Math. Z. 89 (1965) 345–347. [24] D.I. Zaitsev, Solvable groups of finite rank, in: Groups with Restricted Subgroups, “Naukova Dumka”, Kiev, 1971, pp. 115–130 (in Russian).
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Journal of Algebra www.elsevier.com/locate/jalgebra
On groups whose factor-group modulo the hypercentre has finite section p-rank ✩ M.R. Dixon a , L.A. Kurdachenko b , J. Otal c,∗ a
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA b Department of Algebra, National University of Dnepropetrovsk, 49010 Dnepropetrovsk, Ukraine c Departamento de Matemáticas-IUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain
a r t i c l e
i n f o
Article history: Received 15 January 2015 Available online xxxx Communicated by E.I. Khukhro MSC: 20F14 20F19
a b s t r a c t Several theorems of the following type are obtained: Let G be a locally generalized radical group and let p be a prime. Suppose that G/ζk (G) has finite section p-rank at most r. Then γk+1 (G) has finite section p-rank. Moreover there exists a function τ (r, k) such that rp (γk+1 (G)) ≤ τ (r, k). © 2015 Elsevier Inc. All rights reserved.
Keywords: Finite section p-rank of a group Generalized radical group Hypercentre of a group Locally nilpotent residual of a group
✩ The work of the third author was supported by the Department of I+D of the Government of Aragón (Spain) and FEDER funds from European Union. Some of this work was carried out at the University of Napoli, from which the first two authors received financial support from STAR grant Napoli-24call2013-24. These authors would like to thank the university and its faculty and students for their hospitality during that time. * Corresponding author. E-mail addresses: [email protected] (M.R. Dixon), [email protected] (L.A. Kurdachenko), [email protected] (J. Otal).
http://dx.doi.org/10.1016/j.jalgebra.2015.06.005 0021-8693/© 2015 Elsevier Inc. All rights reserved.
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Introduction It is well-known that if the kth term, ζk (G), of the upper central series of a group G coincides with G, then the (k + 1)th term, γk+1 (G), of the lower central series of G is trivial. R. Baer [1] obtained a nice generalization of this result proving that if the factor group G/ζk (G) is finite, then γk+1 (G) is also finite. This result was extended, using the broad general result [19, Theorem 4.21] and a theorem of Schlette [21], to the class of Chernikov groups, a result stated more explicitly in the paper [18] of J. Otal and J.M. Peña. The class of Chernikov groups is a subclass of the class of groups of finite (special) rank. Here a group G has finite rank r(G) = r if every finitely generated subgroup of G can be generated by r elements and r is the least positive integer with this property. This concept (and the term special rank itself) was introduced by A.I. Maltsev [16]. This result concerning Chernikov groups was itself recently extended to the class of groups of finite rank: in the paper [11] the authors showed that if G/ζk (G) is a locally generalized radical group of finite rank, then γk+1 (G) also has finite rank. We recall that a group G is called generalized radical if it has an ascending series whose factors are either locally nilpotent or locally finite. Accordingly, a group G is locally generalized radical if every finitely generated subgroup of G is generalized radical. We note that the class of locally generalized radical groups is very broad containing, in particular, the classes of locally finite groups and locally soluble groups. Unlike the result of [21], the result of [11] does not hold for all groups since the paper [14] contains an example of a group G whose central factor group is of finite rank but whose derived subgroup is of infinite rank. A further important numerical invariant of a group is its section p-rank, for the prime p, and it is this invariant that concerns us most here. For the prime p we say that a group G has finite section p-rank rp (G) = r if every elementary abelian p-section of G is finite of order at most pr and there is an elementary abelian p-section A/B of G such that |A/B| = pr . If the orders of the elementary abelian p-sections of G are not bounded, then we say that G has infinite section p-rank, and if G has no elementary abelian p-sections, then we will say that rp (G) = 0. Clearly, the concept of section p-rank generalizes the concept of rank. Indeed, if a group G has finite rank r, then G has finite section p-rank for every prime p and rp (G) ≤ r. It is also clear that if G is a torsion-free abelian group of section p-rank at most t, then G has rank at most t. Furthermore, the property of having finite section p-rank is preserved under taking subgroups, factor groups and extensions. Let P be the set of all primes and let σ : P −→ N0 be a function. We shall say that the group G has finite section rank σ if rp (G) = σ(p) for each prime p. Following D.J.S. Robinson [19, Section 9.3], we shall say that a group G has finite abelian section rank if every elementary abelian p-section of G is finite. It is a well-known fact that if a locally generalized radical group has finite abelian section rank, then rp (G) is finite for each prime p. Our first main result is as follows.
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Theorem A. Let G be a locally generalized radical group and let p be a prime. Suppose that G/ζk (G) has finite section p-rank at most r. Then γk+1 (G) has finite section p-rank. Moreover, there exists a function τ (r, k) such that rp (γk+1 (G)) ≤ τ (r, k). The case k = 1 here is [3, Theorem A]. There are a number of interesting consequences of this result which we now list and which mostly follow immediately. Corollary A1. Let G be a locally generalized radical group. Suppose that G/ζk (G) has finite section rank σ, for some function σ : P −→ N0 . Then γk+1 (G) has finite section rank and rp (γk+1 (G)) ≤ τ (σ(p), k), for every prime p. Corollary A2. Let G be a locally generalized radical group. Suppose that G/ζk (G) has finite abelian section rank. Then γk+1 (G) has finite abelian section rank. Corollary A3. Let G be a locally generalized radical group. Suppose that G/ζk (G) has finite rank r. Then γk+1 (G) has finite rank, and there is a function τ1 (r, k) such that r(γk+1 (G)) ≤ τ1 (r, k). Corollary A4. Let G be a finite group. Suppose that G/ζk (G) has rank r. Then there exists a function τ1 (r, k) such that r(γk+1 (G)) ≤ τ1 (r, k). Suppose that X is a class of groups. For the group G we let GX = ∩{H G|G/H ∈ X}, / X, but in many important classes of groups the X-residual of G. In general G/GX ∈ membership does hold. We shall let GN denote the nilpotent residual of a group G and let GLN denote the locally nilpotent residual of G. In the paper [6], Baer’s theorem was generalized in a different direction. There the authors proved that if G is a group whose hypercentre has finite index, then the locally nilpotent residual, GLN , of G is finite and G/GLN is hypercentral. In [13] a quantitative version of this result was given. Here we provide a substantial generalization of Baer’s theorem. We shall prove: Theorem B. Let G be a group and let p be a prime. Suppose that the upper hypercentre of G contains a G-invariant subgroup Z such that G/Z is locally finite and has finite section p-rank r. Then the locally nilpotent residual, GLN , of G is locally finite, Π(GLN ) ⊆ Π(G/Z), and there is a function τ2 (r) such that rp (GLN ) ≤ τ2 (r). Moreover, G/GLN is locally nilpotent. The following consequences can be deduced. We observe that the groups in the question are generalized radical groups.
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Corollary B1. Let G be a group. Suppose that the hypercentre of G contains a G-invariant subgroup Z such that G/Z is locally finite and has finite section rank σ, for some function σ : P −→ N0 . Then GLN is locally finite, Π(GLN ) ⊆ Π(G/Z) and rp (GLN ) ≤ τ2 (σ(p)), for all primes p. Moreover, G/GLN is hypercentral. This particular corollary can be rephrased as follows. Corollary B2. Let G be a group. Suppose that the hypercentre of G contains a G-invariant subgroup Z such that G/Z is locally finite and has finite abelian section rank. Then GLN is locally finite and has finite abelian section rank. Moreover, G/GLN is hypercentral. Corollary B3. Let G be a group. Suppose that the hypercentre of G contains a G-invariant subgroup Z such that G/Z is locally finite and has finite rank r. Then GLN is locally finite, Π(GLN ) ⊆ Π(G/Z), and there exists a function τ3 such that r(GLN ) ≤ τ3 (r). Moreover, G/GLN is hypercentral. Our next result has also appeared in [10] for periodic groups. Corollary B4. Let G be a group. Suppose that the hypercentre of G contains a G-invariant subgroup Z such that G/Z is a Chernikov group. Then GLN is also Chernikov. Furthermore, G/GLN is hypercentral. The condition that G/Z is locally finite in Theorem B and its corollaries is essential. In Section 4 we give an example to show that the result breaks down completely in the absence of this hypothesis. Finally, A. Mann [17] proved that if G/ζ(G) is locally finite of exponent e, then γ2 (G) is also locally finite of finite exponent m(e), for some function m. This result was generalized in the paper [12], where it is proved that if G/ζk (G) is locally finite of exponent e, then γk+1 (G) is also locally finite of finite exponent. Our final main result gives yet another generalization of Mann’s theorem. Theorem C. Let G be a group. Suppose that the hypercentre of G contains a G-invariant subgroup Z such that G/Z is locally finite and has exponent e. Then the locally nilpotent residual, GLN , of G is locally finite of exponent at most β2 (e), for some function β2 depending upon e only. Our notation is standard. We let ζ∞ (G) denote the hypercentre of the group G. 1. Proof of Theorem A In this section we shall provide the proof of Theorem A and first briefly discuss the class of locally generalized radical groups, which has been studied in several recent papers. A result that is fundamental in much of what follows concerns the structure of
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such groups with finite section p-rank at most r. For convenience we include this result here and note that the proof occurs in both [3] and [5]. Theorem 1.1. Let G be a locally generalized radical group and let p be a prime. If every finitely generated subgroup of G has finite section p-rank at most r, then G has normal subgroups 1 ≤ T ≤ L ≤ K ≤ S ≤ G such that (i) (ii) (iii) (iv)
T is locally finite and G/T is soluble-by-finite of finite rank; L/T is a torsion-free nilpotent group of rank at most r; K/L is a finitely generated torsion-free abelian group of rank at most r; G/K is finite of order at most θ1 (r) and S/T is the soluble radical of G/T of derived length at most f1 (r).
Furthermore, the Sylow p-subgroups of T are Černikov of finite rank at most f2 (r), for some function f2 : N −→ N. It follows easily from this theorem that the group G/T has rank at most θ2 (r) = 2r +log2 (θ1 (r)). Of course we have no control over the p -subgroups of T in this theorem. It was also shown in [5, Theorem D] that if G is a locally generalized radical group and if there is an integer r such that r(A) ≤ r for all abelian subgroups A, then G has finite rank at most f (r) + r(5r + 1)/2 + 1, for some function f of r only. It is easy to extract the following result from the proof of [5, Theorem D]. Lemma 1.2. Let G be a locally finite group and suppose that the abelian subgroups of G have rank at most r. Then G has rank at most r(5r + 1)/2 + 1. With these preliminaries taken care of we now show how to control the structure of certain factor groups. Next we quote the following result whose proof appears in [3, Lemma 3.10]. Lemma 1.3. Let G be a group, let p be a prime and let A be a normal abelian p-subgroup of G. Suppose that G satisfies the following conditions: (i) G/CG (A) is a periodic abelian p -group; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank r. Then G/CG (A) is finite of rank at most r. We shall prove certain variations of this type of result, the first one being as follows. Lemma 1.4. Let G be a group, let p be a prime and let A be a normal abelian p-subgroup of G. Suppose that G satisfies the following conditions:
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(i) G/CG (A) is a locally finite group of finite section p-rank t; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank r. Then G/CG (A) is Chernikov of rank at most t + r(5r + 1)/2 + 1, and if t = 0 then G/CG (A) is finite. Proof. Let V = G/CG (A). If Q is a p -subgroup of V then, by Lemma 1.3, the abelian subgroups of Q are finite of rank at most r. By a theorem of Hall and Kulatilaka [7], every infinite locally finite group contains an infinite abelian subgroup, so Q must be finite. In particular, if t = 0, then V must be finite. If T is a Sylow p-subgroup of V , then T has finite section p-rank at most t and hence T has rank at most t by [3, Corollary 2.3]. Therefore, T is a Chernikov group by [19, Corollary 2]. It follows that V is a locally finite group with Chernikov Sylow q-subgroups for all primes q and a theorem of Belyaev (see [4, Theorem 3.5.15]) implies that V is almost locally soluble. Hence V /Op (V ) is a Chernikov group (see [4, Theorem 2.5.12]). Since Op (V ) is finite, by our work above, V is Chernikov. Finally, let F be an arbitrary finite subgroup of V . If S is a Sylow q-subgroup of F , for some prime q = p, then S is finite and has rank at most ρ(r) = r(5r + 1)/2, by [20, Corollary 5], so S has at most ρ(r) generators. If S is a Sylow p-subgroup of F , then S has rank at most t, so S has at most t-generators. By [15, Theorem 1], it follows that F has at most max(t, ρ(r)) + 1 generators. We deduce that G/CG (A) has rank at most t + ρ(r) + 1, as required. 2 Our next result shows that the property of having bounded section p-rank is really a local property. Lemma 1.5. Let G be a group, let L be a local system for G and let p be a prime. If there exists a positive integer t such that the section p-rank of every subgroup H ∈ L is at most t, then G has section p-rank at most t. Proof. Let U, V be subgroups of G such that U V and V /U is a finite elementary abelian p-group. Then V /U = a1 U × . . . × an U , for certain elements a1 , a2 , . . . , an of V such that api ∈ U for each i, where 1 ≤ i ≤ n. Since L is a local system, there is subgroup Ki ∈ L such that ai ∈ Ki , for 1 ≤ i ≤ n and, for the same reason, there is a subgroup Y ∈ L such that Ki ≤ Y , for 1 ≤ i ≤ n. Then A = a1 , . . . , an ≤ Y . Now V = AU so V /U ∼ = A/A ∩ U . Since Y ∈ L, it has section p-rank at most t and, since A ≤ Y , it follows that the elementary abelian p-sections of A have rank at most t. Hence n ≤ t and it follows that rp (G) ≤ t. 2
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We now start to transfer hypotheses concerning factor groups to conclusions about commutators. Lemma 1.6. Let G be a group, let p be a prime and let A be a normal abelian p-subgroup of G. Suppose that G satisfies the following conditions: (i) G/CG (A) = x1 CG (A), . . . , xm CG (A) , where x1 , . . . , xm ∈ G; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank t. Then [A, G] has finite section p-rank at most tm. Proof. Let g be an arbitrary element of G and let ζg : A −→ [A, g] be the endomorphism defined by ζg (a) = [a, g]. Since B ≤ ζ(G) ∩ A ≤ CA (g) = ker(ζg ) it follows that A/CA (g) has finite section p-rank at most t. However A/CA (g) ∼ = [A, g], so rp ([A, g]) ≤ t also. By [11, Lemma 1.1] we have [A, G] = [A, x1 ] . . . [A, xm ] and it then follows easily that rp ([A, G]) ≤ tm, as required. 2 This yields the following result, which is analogous to [11, Lemma 1.2]. Lemma 1.7. Let G be a group, let p be a prime and let A be a normal abelian p-subgroup of G. Suppose that G satisfies the following conditions: (i) G/CG (A) is a group of finite rank t; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank r. Then [A, G] has finite section p-rank at most tr. Proof. Let L be the local family consisting of all the finitely generated subgroups of G/CG (A). If F/CG (A) ∈ L then F/CG (A) = x1 CG (A), . . . , xm CG (A) , for certain elements x1 , . . . , xm of G such that m ≤ t. Lemma 1.6 implies that [A, F ] has finite section p-rank at most rm ≤ rt. Of course {[A, F ]|F/CG (A) ∈ L} is a local system for [A, G] and Lemma 1.5 implies that rp ([A, G]) ≤ tr. 2 We require further information concerning normal abelian subgroups and, for our next result, we recall that if T is a periodic subgroup of GLn (Q), then T is finite and there is a function ρ, of n only, such that |T | ≤ ρ(n) (see [22, Theorem 9.33], for example). From this it follows that a periodic group of automorphisms of a torsion-free abelian group of finite Z-rank n is finite of order at most ρ(n). Lemma 1.8. Let G be a group, let p be a prime and let A be a normal torsion-free abelian subgroup of G. Suppose that
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(i) G/CG (A) is periodic; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank r. Then G/CG (A) is finite and has order at most ρ(r). Proof. Let T /B be the torsion subgroup of A/B and let a ∈ T , so there is a positive integer k such that ak ∈ B. If g ∈ G then (ag )k = (ak )g = ak . Since A is torsion-free, it follows that ag = a and hence T ≤ ζ(G). Now, A/T is torsion-free and hence has rank at most r, by our hypotheses. Let U = CG (A/T ) and note that, by our remark above, G/U has order at most ρ(r). If a ∈ A, u ∈ U , then au = ac, for some element c ∈ T . Since G/CG (A) is periodic, there is a positive integer m such that um ∈ CG (A), so a = u−m aum = acm . It follows that cm = 1 and, since A is torsion-free, we deduce that c = 1. Hence U = CG (A/T ) = CG (A) and the result now follows. 2 In our next result we require structural information about locally generalized radical groups, as elucidated in Theorem 1.1. Lemma 1.9. Let G be a locally generalized radical group, let p be a prime and let A be an abelian normal subgroup of G. Suppose that (i) G/CG (A) has finite section p-rank t; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank r; (iii) The torsion subgroup of A is a p-group. Then [A, G] has finite rank and there is a function θ4 such that r([A, G]) ≤ θ4 (t, r). Proof. Let Q/B be the Sylow p -subgroup of A/B and let a ∈ Q. Then there is a p -number k such that ak ∈ B. If g ∈ G, then (ag )k = (ak )g = ak and, since A is abelian, we have (a−1 ag )k = 1. However, the hypothesis on A implies that ag = a so we deduce that Q ≤ ζ(G). The choice of Q implies that A/Q contains no p -elements. Since A/Q has p-rank at most r, [3, Lemma 2.1] shows that A/Q has finite rank at most r. Let T be the torsion subgroup of A and let R/CG (A) be the largest normal torsion subgroup of G/CG (A). By Lemma 1.4, R/CR (T ) is a Chernikov group of rank at most t + r(5r + 1)/2 + 1. Furthermore, Lemma 1.8 shows that R/CR (A/T ) is finite of order at most ρ(r). By Remak’s theorem, we have the embedding R/(CR (T ) ∩ CR (A/T )) −→ R/CR (T ) × R/CR (A/T ), which shows that R/(CR (T ) ∩ CR (A/T )) is Chernikov of rank at most t + r(5r + 1)/2 + 1 + log2 (ρ(r)). Now let a ∈ A and x ∈ CR (T ) ∩ CR (A/T ). Then ax = ac for some c ∈ T . Since T is a p-group, there is a p-number m such that cm = 1. Then x−m axm = acm = a so xm ∈ CG (A). Hence CR (T ) ∩ CR (A/T )/CG (A) is a p-group. By our hypotheses and
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[3, Corollary 2.3] we deduce that CR (T ) ∩ CR (A/T )/CG (A) is Chernikov of rank at most t. Hence R/CG (A) is Chernikov of rank at most 2t + r(5r + 1)/2 + 1 + log2 (ρ(r)). By Theorem 1.1, G/R has rank at most θ2 (t) and hence G/CG (A) has rank at most θ2 (t) + 2t + r(5r + 1)/2 + 1 + log2 (ρ(r)) = θ3 (t, r). Applying [11, Lemma 1.2], we deduce that [A, G] has finite rank at most rθ3 (t, r) = θ4 (t, r), as required. 2 Proposition 1.10. Let G be a locally generalized radical group, let p be a prime and let A be a normal abelian subgroup of G. Suppose that G satisfies the following conditions: (i) G/CG (A) has finite section p-rank t; (ii) ζ(G) ∩ A contains a subgroup B such that A/B has finite section p-rank r. Then [A, G] has finite section p-rank at most θ4 (t, r). Proof. Let Q be the Sylow p -subgroup of A. By Lemma 1.9, [A/Q, G/Q] = V /Q has finite rank at most θ4 (t, r) and hence V also has finite section p-rank at most θ4 (t, r). However, [A, G] ≤ V so the result follows. 2 Proof of Theorem A. Let 1 = Z0 ≤ Z1 ≤ . . . ≤ Zk−1 ≤ Zk = Z be the upper central series of G. We use induction on k. If k = 1, then the central factor group G/Z1 has finite section p-rank r and [3, Theorem A] implies that γ2 (G) has finite section p-rank and that there is a function λ2 such that rp (γ2 (G)) ≤ λ2 (r). Assume now that k ≥ 1 and apply our induction hypothesis to G/Z1 , so that there is a function τ such that rp (γk (G/Z1 )) ≤ τ (r, k − 1). Set K/Z1 = γk (G/Z1 ), let L = γk (G) and note that K = LZ1 . Then K = [LZ1 , LZ1 ] = L , since Z1 = Z(G). Also, [K, G] = [L, G] = γk+1 (G). Theorem A of [3] shows that L has finite section p-rank and rp (L ) ≤ λ2 (τ (r, k − 1)). Well-known properties of the upper and lower central series show that [L, Z] = 1, so G/CG (L) is an image of G/Z and hence rp (G/CG (L)) ≤ r. We wish to apply Proposition 1.10 to G/L . To this end we note that L/L ∩Z1 L /L ≤ L/L ∩ ζ(G/L ) and that (L/L )/(L/L ∩ Z1 L /L ) ∼ = (L/L )(Z1 L /L )/(Z1 L /L ) ∼ = LZ1 /L Z1 ∼ = L/(L ∩ Z1 L ), an image of L/L ∩ Z1 . Certainly, rp (L/L ∩ Z1 ) = rp (LZ1 /Z1 ) ≤ rp (K/Z1 ) ≤ τ (r, k − 1). Also, (G/L )/CG/L (L/L ) is an image of G/CG (L), so rp ((G/L )/CG/L (L/L )) ≤ r.
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Then, by Proposition 1.10, [L/L , G/L ] = [L, G]/L = γk+1 (G)/L has finite section p-rank at most θ4 (r, τ (r, k − 1)). It follows that γk+1 (G) has finite section p-rank and that rp (γk+1 (G)) ≤ λ2 (τ (r, k − 1)) + θ4 (r, τ (r, k − 1)) = τ (r, k).
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Proposition 1.11. Let G be a locally generalized radical group. If there exists a positive integer r such that rp (G) ≤ r, for each prime p, then G has finite rank at most 4r + 2. Proof. By Theorem 1.1, G has a series 1 ≤ T ≤ L ≤ K ≤ S ≤ G. The group L/T has rank at most r, by [24] and K/L is also of rank at most r. Hence K/T has rank at most 2r. We note that T is locally finite. Let F be a finite subgroup of T , let q ∈ Π(F ) and let S be a Sylow q-subgroup of F . Then [3, Lemma 2.2] shows that S has rank at most r, so that S has at most r generators. Since this is true for each prime q, it follows from [15, Theorem 1] that F has at most r + 1 generators. Hence T has rank at most r + 1. The same argument applies in the finite factor group G/K, so that it has rank at most r + 1 too. Hence G has rank at most 2(r + 1) + 2r = 4r + 2, as required. 2 Proof of Corollary A3. Suppose that G/ζk (G) has finite rank r. As we saw above, this means that rp (G/ζk (G)) ≤ r, for each prime p. Theorem A then implies that rp (γk+1 (G)) ≤ τ (r, k), for each prime p. By Proposition 1.11, γk+1 (G) has rank at most 4τ (r, k) + 2 = τ1 (r, k). 2 2. Proof of Theorem B Let G be a group whose centre contains a subgroup Z such that G/Z is finite of order t. Then |G/ζ(G)| ≤ t and hence, by a result of J. Wiegold [23], we have |G | ≤ w(t) = tm , where m = (1/2)(logp t − 1) and p is the least prime dividing t. If t1 = |G/ζ(G)| then t1 is a divisor of t and it follows that Π(G/ζ(G)) ⊆ Π(G/Z). It follows from [2, Zusatz] that the orders of the elements of G divide t21 and hence divide t2 . Hence Π(G ) ⊆ Π(G/Z). If α is a real number then we let ι(α) denote the least integer not less than α. Lemma 2.1. Let p be a prime, let G be a finite group, let Z be a G-invariant subgroup of the hypercentre of G and let GN be the nilpotent residual of G. Then Π(GN ) ⊆ Π(G/Z) and there exist functions τ2 , τ3 such that rp (GN ) ≤ τ2 (rp (G/Z)) and |GN | ≤ τ3 (|G/Z|). Proof. Let |G/Z| = t, r = rp (G/Z) and π = Π(G/Z). There is a series of G-invariant subgroups 1 = Z0 ≤ Z1 ≤ . . . ≤ Zn ≤ Zn+1 = Z
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whose factors are G-central. Let C = CG (Z). By a theorem of L.A. Kaluzhnin [9], G/C is nilpotent and clearly Z ≤ CG (C), so that |G/CG (C)| ≤ t. We have C/Z ∩ C ∼ = CZ/Z, so C/C ∩Z has order dividing t, rp (C/Z ∩C) ≤ r and Π(C/Z ∩C) ⊆ π. Since C ∩Z ≤ ζ(C), the remark preceding the lemma shows that |C | ≤ w(t) and Π(C ) ⊆ π. Applying [3, Proposition 3.3] to C shows that there exists a function λ such that rp (C ) ≤ λ(r) = r(3r + 1)/2 + r2 ι(log2 r). Since G/C is nilpotent, so is (G/C )/CG/C (C/C ) and [11, Corollary 2.5] shows that C/C has a Z-decomposition, say ∞ ∞ C/C = ζZG (C/C ) ⊕ EZG (C/C ). ∞ Clearly (C ∩ Z)C /C ≤ ζZG (C/C ) from which it follows that the order of R/C = ∞ EZG (C/C ) is a divisor of t, rp (R/C ) ≤ r and Π(R/C ) ⊆ π. Furthermore, the factor C/R ∼ = (C/C )/(R/C ) is G-hypercentral and hence the hypercentre of G/R contains C/R. Since G/C is nilpotent, G/R is therefore nilpotent and hence GN ≤ R. Finally,
|R| = |C | · |R/C | ≤ w(t)t = tm+1 = τ3 (t), rp (R) ≤ r + λ(r) = τ2 (r) and Π(R) ⊆ π. The result follows. 2 If G is a group, then we let Tor(G) denote the maximal normal periodic subgroup of G and note that, when G is locally nilpotent, Tor(G) is a characteristic subgroup of G such that G/Tor(G) is torsion-free. Before proving the next result we make the following observation. Suppose that G is a finitely generated group and suppose that Z is a G-invariant subgroup of the hypercentre of G such that G/Z is finite. Then ζ∞ (G) is also finitely generated, and locally nilpotent. Hence ζ∞ (G) has the maximum condition and it follows that there is a positive integer n such that ζ∞ (G) = ζn (G). Baer’s Theorem [1] implies that γn+1 (G) is finite. Hence GN is finite and G/GN is nilpotent. Lemma 2.2. Let p be a prime, let G be a finitely generated group, and let Z be a G-invariant subgroup of the hypercentre of G. Suppose that G/Z is finite. Then GN is finite and Π(GN ) ⊆ Π(G/Z). Also, rp (GN ) ≤ τ2 (rp (G/Z)) and |GN | ≤ τ3 (|G/Z|). Furthermore, G/GN is nilpotent. Proof. Let |G/Z| = t, r = rp (G/Z) and π = Π(G/Z). As we remarked above, the nilpotent residual GN is finite and G/GN is nilpotent. Since G/GN is finitely generated, its torsion subgroup K/GN is finite and hence G/K is a finitely generated torsion-free nilpotent group. Since Z is a finitely generated nilpotent subgroup, its torsion subgroup T is finite. By [8], Z contains a normal torsion-free subgroup U such that the orders of the elements
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of Z/U are the divisors of some positive integer k. Let V = Z k ≤ U , so that V is also torsion-free. Clearly, V is G-invariant and G/V is periodic. However, G/V is finitely generated and nilpotent-by-finite, so G/V must be finite. The hypercentre of G/V contains the G-invariant subgroup ZV /V . Furthermore, |(G/V )/(ZV /V )| ≤ t, rp ((G/V )/(ZV /V )) ≤ r and Π((G/V )/(ZV /V )) ⊆ π. Let D/V be the nilpotent residual of the finite group G/V . By Lemma 2.1, |D/V | ≤ τ3 (t), rp (D/V ) ≤ τ2 (r) and Π(D/V ) ⊆ π. Since G/K and G/D are nilpotent, we have that G/(K ∩ D) is also nilpotent and hence GN ≤ K ∩ D. Furthermore, V is torsion-free and K is finite, so V ∩ K = 1. Then K ∩D ∼ = (K ∩ D)V /V ≤ D/V. It follows that |K ∩ D| ≤ τ3 (t), rp (K ∩ D) ≤ τ2 (r) and Π(K ∩ D) ⊆ π. Since GN ≤ K ∩ D this completes the proof. 2 Proof of Theorem B. Let L be the family of all finitely generated subgroups of G. If U ∈ L, then the hypercentre of U contains the U -invariant subgroup U ∩ Z = ZU and U/ZU is a finitely generated, locally finite group. Hence U/ZU is finite. Also, since U/ZU ∼ = U Z/Z ≤ G/Z, we have rp (U/ZU ) ≤ r and Π(U/ZU ) ⊆ π. Lemma 2.2 shows that U N is finite, rp (U N ) ≤ τ2 (r) and Π(U N ) ⊆ π. Furthermore, U/U N is nilpotent. Now let V be another subgroup in L. Then there exists W ∈ L such that U, V ≤ W . Of course, W N ∩ U U and U/(W N ∩ U ) is nilpotent. Hence U N ≤ W N and, similarly, V N ≤ W N . It follows that R = U ∈L U N is a locally finite subgroup of G such that Π(R) ⊆ π. By Lemma 1.5, rp (R) ≤ τ2 (r). Also, R is a normal subgroup of G, since U N is a normal subgroup of U for each U ∈ L. Let F/R be a finitely generated subgroup of G/R. Then there is a finitely generated subgroup V of G such that F = V R. Of course, V N ≤ R ∩ V and hence V /R ∩ V ∼ = V R/R = F/R is nilpotent. Consequently, G/R is locally nilpotent and it follows that GLN ≤ R. Clearly, Π(GLN ) ⊆ π and rp (GLN ) ≤ τ2 (r). Next, we let Y be a normal subgroup of G such that G/Y is locally nilpotent. If U ∈ L, then U Y /Y is nilpotent and hence U N ≤ U ∩ Y . In particular U N ≤ Y , and we have R = U ∈L U N ≤ Y . Since this is true for all such normal subgroups Y , we have R ≤ GLN and hence R = GLN . Hence G/GLN is locally nilpotent. This completes the proof. 2 Proof of Corollaries B1–B4. For Corollary B1 we suppose that G has finite section rank σ. By Theorem B, G/GLN is locally nilpotent. Let K/GLN be the hypercentre
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of G/GLN . Then G/K is locally nilpotent. However, ZGLN /GLN ≤ K/GLN , so G/K is also locally finite and has finite section rank. The p-component of G/K is Chernikov and hence hypercentral, using [3, Corollary 2.3] and well-known facts. It follows that G/K is hypercentral, so that G = K. Hence Corollaries B1 and B2 follow. For Corollary B3, it follows from Corollary B1 that rp (GLN ) ≤ τ2 (r), for all primes p. It follows from Proposition 1.11 that GLN is of rank at most τ3 (r) = 4τ2 (r) + 2. For Corollary B4, we note that GLN has finite rank, by Corollary B3, and that there are only finitely many primes in Π(GLN ). Hence G is Chernikov. 2 3. Proof of Theorem C Let L be the family of all finitely generated subgroups of G. If U ∈ L, then the hypercentre of U contains the U -invariant subgroup U ∩ Z = ZU . Furthermore, U/ZU is finitely generated and locally finite. Hence U/ZU is finite and, since U/ZU ∼ = U Z/Z ≤ G/Z, the exponent of U/ZU is at most e. By our remark before the proof of Lemma 2.2, the nilpotent residual U N of U is finite and, by [12, Theorem B], U N has finite exponent β2 (e), for some function β2 of e only. Now let V be another subgroup in L. Then there exists W ∈ L such that U, V ≤ W . Of course, W N ∩ U U and U/(W N ∩ U ) is nilpotent. Hence U N ≤ W N and similarly V N ≤ W N . Hence R = U ∈L U N is a locally finite subgroup of G. Since every subgroup of U N has finite exponent at most β2 (e), R also has finite exponent at most β2 (e). As in the proof of Theorem B, R contains GLN . This completes the proof. 4. An example Finally, we give the example promised in the Introduction. Let G = a d be an infinite dihedral group, where a is an element of infinite order, d has order 2 and ad = a−1 . Let A0 = a and, for each n ≥ 0, let An+1 = A2n . Each of the subgroups An is clearly normal in G and G/An is a finite dihedral group of order 2n+1 . In the group Dn = G/An let an = aAn and vn = dAn . Let b be an element of order 2 and let Kn = b Dn , the wreath product of b by Dn . Let Bn denote the base group of this wreath product, so that Kn = Bn Dn . Let B = Drn∈N Bn denote the direct product of the groups Bn , let D = Crn∈N Dn denote the Cartesian product of the groups Dn and define an action of D on B as follows: If b = (bn ) ∈ Drn∈N Bn and y = (dn ) ∈ Crn∈N Dn let by = (bdnn ). Let H = B D be the corresponding semidirect product defined by this action. The mapping f : G −→ D defined by f (x) = (xAn )n∈N is clearly a homomorphism and, since ∩n∈N An = 1, Remak’s theorem implies that this homomorphism is an embedding of G into D. Thus we may form the subgroup of H defined by L = B f (G).
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A short calculation (which we omit) shows that B = ζ∞ (L) and that the length of the upper central series of L is the first infinite ordinal, ω. Hence L/ζ∞ (L) ∼ = f (G) ∼ =G is infinite dihedral, so is of finite rank. We suppose, for a contradiction, that L contains a normal subgroup P such that P has finite rank and L/P is locally nilpotent. Since B is an elementary abelian 2-group and L/B is polycyclic, it follows that P is also polycyclic. If P is periodic, then P is finite, so f (G) ∩ P is a finite normal subgroup of the infinite dihedral group f (G). This implies that f (G) ∩ P = 1 and hence f (G)P/P ∼ = f (G) is locally nilpotent, which is a contradiction. Hence P is non-periodic and, since L has torsion-free rank 1, it follows that P has torsion-free rank 1. Since P is polycyclic, it contains an infinite cyclic normal subgroup of finite index and hence a characteristic such subgroup u . Since L = B ·f (G) and f (G) is dihedral, we have u = xf (as ) for some element x ∈ B and some integer s. Since x ∈ ζk (L), for some natural number k, it is easy to see that, for a suitable positive power l, we have ul = f (asl ). However, ul has infinitely many conjugates in L whereas, by construction, ul is a normal subgroup of L. This gives us the required contradiction and it follows that L has no normal subgroup of finite rank with locally nilpotent factor group. References [1] R. Baer, Representations of groups as quotient groups. I, Trans. Amer. Math. Soc. 58 (1945) 295–347. [2] R. Baer, Endlichkeitskriterien für Kommutatorgruppen, Math. Ann. 124 (1952) 161–177. [3] A. Ballester-Bolinches, S. Camp-Mora, L.A. Kurdachenko, J. Otal, Extension of a Schur theorem to groups with a central factor with a bounded section rank, J. Algebra 393 (2013) 1–15. [4] M.R. Dixon, Sylow Theory, Formations and Fitting Classes in Locally Finite Groups, Ser. Algebra, vol. 2, World Scientific, Singapore, 1994. [5] M.R. Dixon, L.A. Kurdachenko, N.V. Polyakov, Locally generalized radical groups satisfying certain rank conditions, Ric. Mat. 56 (2007) 43–59. [6] M. De Falco, F. de Giovanni, C. Musella, Y.P. Sysak, On the upper central series of infinite groups, Proc. Amer. Math. Soc. 139 (2) (2011) 385–389. [7] P. Hall, C.R. Kulatilaka, A property of locally finite groups, J. London Math. Soc. 39 (1964) 235–239. [8] H. Heineken, L.A. Kurdachenko, Groups with subnormality for all subgroups that are not finitely generated, Ann. Mat. Pura Appl. 169 (1995) 203–232. [9] L.A. Kaluzhnin, Über gewisse Beziehungen zwischen einer Gruppe und ihren Automorphismen, in: Bericht über die Mathematiker-Tagung in Berlin, Januar, 1953, Deutscher Verlag der Wissenschaften, Berlin, 1953, pp. 164–172. [10] L.A. Kurdachenko, J. Otal, Groups with Chernikov factor-group by hypercentral, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM (2015), http://dx.doi.org/10.1007/ s13398-014-0201-7. [11] L.A. Kurdachenko, J. Otal, The rank of the factor-group modulo the hypercenter and the rank of the some hypocenter of a group, Cent. Eur. J. Math. 11 (10) (2013) 1732–1741. [12] L.A. Kurdachenko, J. Otal, A.A. Pypka, Relationships between the factors of the upper and lower central series of a group, Bull. Malaysian Math. Sci. Soc. (2015), in press. [13] L.A. Kurdachenko, J. Otal, I.Ya. Subbotin, On a generalization of Baer theorem, Proc. Amer. Math. Soc. 141 (8) (2013) 2597–2602. [14] L.A. Kurdachenko, P. Shumyatsky, The ranks of central factor and commutator groups, Math. Proc. Cambridge Philos. Soc. 154 (1) (2013) 63–69. [15] A. Lucchini, A bound on the number of generators of a finite group, Arch. Math. (Basel) 53 (4) (1989) 313–317. [16] A.I. Maltsev, On groups of finite rank, Mat. Sb. 22 (1948) 351–352.
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