On a Generalization of Baer Theorem

R. Baer has proved that if the factor-group G/{\zeta}_{n}(G) of a group G by the member {\zeta}_{n}(G) of its upper central series is finite (here n is a positive integer) then the member {\gamma}_{n+1}(G) of the lower central series of G is also finite. In particular, in this case, the nilpotent residual of G is finite. This theorem admits the following simple generalization that has been published very recently by M. de Falco, F. de Giovanni, C. Musella and Ya. P. Sysak:"If the factor-group G/Z of a group G modulo its upper hypercenter Z is finite then G has a finite normal subgroup L such that G/L is hypercentral". In the current article we offer a new simpler very short proof of this theorem and specify it substantially. In fact, we prove that if |G/Z|=t then |L|\leqt^{k}, where k=(1/2)(log_{p}t+1), and p is the least prime divisor of t.


Introduction
One of the important long-standing results in the Theory of Groups is a classical theorem due to I. Schur [8], which establishes a connection between the factor-group G/ζ(G) of a group G modulo its center ζ(G) and the derived subgroup [G, G] of G. It follows from Schur's theorem [8] that if G/ζ(G) is finite then [G, G] is also finite. A natural question related to this result appears here, namely the question regarding the relationship between the orders |G/ζ(G)| and |[G, G]|. J. Wiegold in the paper [9] obtained the following answer to this question. Let G be a group such that |G/ζ(G)| = t is finite. J. Wiegold proved that there exists a function w such that |[G, G]| ≤ w(t). He also was able to obtain for this function the value w(t) = t m where m = 1 2 (log p t − 1) and p is the least prime divisor of t. Later on, J. Wiegold was able to show that this boundary value may be attained if and only if t = p n for some prime p ( [10]). When t has more than one prime divisor, the picture becomes more complicated.
Various generalizations of Schur's theorem can be found in the mathematical literature. One of the most interesting approaches would be studying the properties of the following question: study properties of the factor-group G/ζ(G) such that the derived subgroup [G, G] satisfies the same property. A class of groups X is said to be a Schur class if for every group G such that G/ζ(G) ∈ X the derived subgroup [G, G] also belongs to X. Schur's classes were introduced in the paper [3]. Besides of the obvious examples of the classes of finite and of locally finite groups, the class of polycyclic-by-finite groups and the class of Chernikov groups are also Schur's classes (see, for example, [7,Theorem 3.9]). In this paper [3] other Schur's classes were found as well.
In the paper [1] R. Baer generalized Schur's theorem in a different direction. We recall that the upper central series of a group G is the ascending series given by ζ 1 (G) = ζ(G) is the center of G, and recursively ζ α+1 (G)/ζ α (G) = ζ(G/ζ α (G)) for all ordinals α and ζ λ (G) = µ<λ ζ µ (G) for every limit ordinal λ. The last term ζ ∞ (G) of this series is called the upper hypercenter of G. G itself is called hypercentral if ζ ∞ (G) = G. In general, the length of the upper central series of G is denoted by zl(G). On the other hand, the lower central series of G is the descending series given by γ 2 (G) = [G, G], and recursively γ α+1 (G) = [γ α (G), G] for all ordinals α and γ λ (G) = µ<λ γ µ (G) for every limit ordinal λ.
R. Baer proved that if for some positive integer n the factor-group G/ζ n (G) is finite, then γ n+1 (G) is finite too ( [1]). In particular, in this case the nilpotent residual of G (that is, the intersection of all normal subgroups N of G such that G/N is nilpotent) is finite. Very recently, in the paper [2], M. de Falco, F. de Giovanni, C. Musella and Ya. P. Sysak obtained the following generalization of this result: Theorem A. Let G be a group and let Z be the upper hypercenter of G. If G/Z is finite, then G has a finite normal subgroup L such that G/L is hypercentral.
In Section 2 we provide an elementary proof of this result, which is considerably shorter than the original one.
Just as in the theorem of Schur, the question on finding a relationship between the factor-group G/ζ ∞ (G) and the hypercentral residual of G (the intersection of all normal subgroups N of G such that G/N is hypercentral) appears to be very natural. More specifically, is there a function (dependeding on the order of G/ζ ∞ (G)) that bounds the order of the hypercentral residual of G?. In this note we show that Theorem A can be significantly improved. We prove that the order of the hypercentral residual of G is bounded by a function of the order of G/ζ ∞ (G) and moreover we are able to give an explicit form of this function. Thus the main result of the current note is the following Theorem B. Let G be a group and let Z be the upper hypercenter of G. Suppose that G/Z is finite and put |G/Z| = t. Then G has a finite normal subgroup L such that G/L is hypercentral. Moreover, |L| ≤ t k , where k = 1 2 (log p t + 1) and p is the least prime divisor of t.

A short proof of Theorem A
The proof makes use of an auxiliary result by N. S. Hekster [5,Lemma 2.4].
Lemma 2.1 (HN1986). Let G be a group, K a subgroup of G, and suppose that G = Kζ n (G) for some positive integer n. Then the following properties holds.
Proof of Theorem A. We note that if zl(G) is finite, the result follows from Baer's theorem [1]. Therefore we may suppose that zl(G) is infinite. Let K be a finitely generated subgroup with the property G = ZK. We have that K is nilpotent-byfinite (see [7,Proposition 3.19] for example). Since G/Z is not nilpotent, neither is K. Set r = zl(K) and let C be the upper hypercenter of K. We claim that C = C ∩ Z. For, otherwise CZ/Z = 1 , which means that the upper hypercenter of G/Z is not identity, a contradiction. Then C = C ∩ Z as claimed. By Baer's theorem [1], γ r+1 (K) is finite. It follows that the nilpotent residual L of K is finite. We now consider the local system L consisting of all finitely generated subgroups of G that contains K. Pick V ∈ L and let C V be the upper hypercenter of V . Clearly we have G = ZV and then Since K/L is nilpotent, so is G/ZL. Since the hypercenter of G/L includes ZL/L, G/L has to be hypercentral.

Proof of Theorem B
Let G be a group, R a ring and A an RG-module. We construct the upper RG-central series of A as the ascending chain of submodules for every ordinal α < λ and ζ RG (A/A λ ) = {0}. The last term A λ of this series is called the upper RG-hypercenter of A and will denoted by ζ ∞ RG (A). If A = A λ , then A is said to be RG-hypercentral. Moreover, if λ is finite, then A is said to be RG-nilpotent.
Let B ≤ C be RG-submodules of A. The factor C/B is called G-eccentric if C G (C/B) = G. An RG-submodule C of A is said to be RG-hypereccentric if it has an ascending series of RG-submodules

It is said that the RG-module A has the Z-decomposition if we can express
is the maximal RG-hypereccentric RG-submodule of A (D. I. Zaitsev [11]). We note that, if A has the Z-decomposition, then E ∞ RG (A) includes every RG-hypereccentric RG-submodule and, in particular, it is unique. Indeed, put E = E ∞ RG (A) and let B be a RG-hypereccentric RG-submodule of A. If (B + E)/E is non-zero, then it has a non-zero simple RG-submodule U/E, say. Since (B + E)/E ∼ = B/(B ∩ E), U/E is RG-isomorphic to some simple RG-factor of B and then G = C G (U/E). But (B + E)/E ≤ A/E ∼ = ζ ∞ RG (A) and then G = C G (U/E), a contradiction that shows B ≤ E. Hence E contains the RGhypereccentric RG-submodules of A.
Lemma 3.1. Let G be a finite nilpotent group and let A be a ZG-module. Suppose that the additive group of A is periodic. Then A has the Z-decomposition.
Proof. Since G is finite, A has a local family L consisting of finite ZG-submodules.
If B ∈ L, applying the results of [11], B has the Z-decomposition. Pick now C ∈ L such that B ≤ C. Then we have Proof. The subgroup Z has a series of G-invariant subgroups 1 = Z 0 ≤ Z 1 ≤ · · · ≤ Z n ≤ Z n+1 = Z whose factors Z j+1 /Z j are G-central. Applying a result due to L. A. Kaloujnine [6], the factor-group G/C G (Z) is nilpotent. Put C = C G (Z) so that Z ≤ C G (C). In particular, |G/C G (C)| ≤ t. Clearly C ∩ Z ≤ ζ(C) and so C/(Z ∩ C) ∼ = CZ/Z is a finite group of order at most t. By Wiegold's theorem [9], the derived subgroup D = [C, C] has order at most w(t). We note that D is G-invariant and C/D is abelian. By the facts proved above, the factor-group (G/D)/C G/D (C/D) is nilpotent. By Lemma 3.1, the ZG-module C/D has the Z-decomposition, that is C/D = ζ ∞ ZG (C/D) E ∞ RG (C/D). Clearly, (C ∩ Z)D/D ≤ ζ ∞ ZG (C/D) and then L/D = E ∞ ZG (C/D) has order at most t. Hence (C/D)/(L/D) is ZG-hypercentral. In other words, the hypercenter of G/L contains C/L. Since G/C is nilpotent so is G/L. Finally, |L| = |D||L/D| ≤ tw(t) = tt m = t m+1 , where m = 1 2 (log p t − 1) and p is the least prime divisor of t, so that m + 1 = 1 2 (log p t − 1) + 1 = 1 2 (log p t + 1). Therefore, it suffices to put f 1 (t) = t k , where k = 1 2 (log p t + 1) and p is the least prime divisor of t.
If G is a group, we denote by T or(G) the maximal periodic normal subgroup of G. T or(G) is a characteristic subgroup of G and, if G is locally nilpotent, G/T or(G) is torsion-free. Lemma 3.3. Let G be a finitely generated group and Z a G-invariant subgroup of the hypercenter of G. Suppose that |G/Z| = t is finite. Then G has a finite normal subgroup L such that G/L is nilpotent. Moreover, |L| ≤ f 1 (t).
Proof. Since G/Z is finite, Z is finitely generated. It follows that Z is nilpotent. Moreover, zl(G) is finite. By Baer's theorem [1], G has a finite normal subgroup F such that G/F is nilpotent. Being finitely generated, G/F has a finite periodic part T or(G/F ) = K/F . As we remarked above, the factor-group (G/F )/(K/F ) ∼ = G/K = B is torsion-free and nilpotent. We have that the subgroup Z is nilpotent and T = T or(G) is finite. Then Z has a torsion-free normal subgroup U such that the orders of the elements of Z/U are the divisors of some positive integer k (see [4,Proposition 2] for example). Put V = Z k so that V ≤ U and V is also torsion-free. By construction, V is G-invariant and G/V is periodic. Being finitely generated nilpotent-by-finite, C = G/V is finite. By Lemma 3.2, the nilpotent residual D of C has order at most f 1 (t).
Clearly V ∩ K = 1 . Applying Remak's theorem, we obtain an embedding G ≤ G/V × G/K = C × B = H. Since B is torsion-free nilpotent, the nilpotent residual of H is exactly D. It follows that G/(G∩D) ∼ = GD/D ≤ H/D is nilpotent. This shows that G ∩ D includes the nilpotent residual L of G. In particular, L is finite and moreover |L| ≤ |G ≤ D| ≤ |D| ≤ f 1 (t).
We are now in a position to show the main result of this paper Proof of Theorem B. Since G/Z is finite, there exists a finitely generated subgroup K such that G = KZ. We pick the family Σ of all finitely generated subgroups of G that contains K. Clearly G is F C-hypercentral and then every finitely generated subgroup of G is nilpotent-by-finite (see [7,Proposition 3.19] for example). If U ∈ Σ, then the hypercenter of U includes a U -invariant subgroup U ∩ Z = Z U such that U/Z U is nilpotent and has order at most t. By Lemma 3.3, U has a finite normal subgroup H U such that U/H U is nilpotent and |H U | ≤ f 1 (t). Being finite-by-nilpotent, the nilpotent residual L U of U is finite and L U has order at most f 1 (t).
Pick Y ∈ Σ such that |L Y | is maximal and let Σ 1 be the family of all finitely generated subgroups of G that contains Y . Pick U ∈ Σ 1 . Then Y ≤ U . The factorgroup U/L U is nilpotent and, since Y /(Y ∩ L U ) ∼ = Y L U /L U ≤ U/L U , Y /(Y ∩ L U ) is nilpotent. It follows that L Y ≤ Y ∩ L U and then L Y ≤ L U . But |L Y | is maximal, so that L U = L Y . In particular, L Y is normal in U for every U ∈ Σ 1 . Then L Y is normal in U∈Σ1 U = G and U/L Y is nilpotent. Thus G/L Y has a local family of nilpotent subgroups, that is G/L Y is locally nilpotent. Then (G/L Y )/(ZL Y /L Y ) is nilpotent since it is finite. It follows that G/L Y is hypercentral, because the upper hypercenter of G/L Y includes ZL Y /L Y .