RACSAM DOI 10.1007/s13398-014-0201-7 ORIGINAL PAPER

Groups with Chernikov factor-group by hypercentral Leonid A. Kurdachenko · Javier Otal

Received: 18 May 2014 / Accepted: 23 October 2014 © Springer-Verlag Italia 2014

Abstract In this paper we extend some classical theorems involving the terms and the factors groups of the central series of a group. Specifically we show that a periodic hypercentralby-Chernikov group is Chernikov-by-hypercentral and obtain explicit bounds that describe numerical invariants of the second structure of the group as function of the first one. Keywords Upper central series of a group · Hypercenter of a group · Hypercentral group · Locally nilpotent residual of a group · p-rank of an abelian group · p-minimax and minimax rank of a group · Chernikov group · Schur’s theorem · Baer’s theorem · Z -decomposition · RG-hypercentral module · RG-nilpotent module Mathematics Subject Classification

20F14 · 20F19 · 20F99

1 Introduction The aim of this paper is the study the relationship between the factor-group of a given group by a normal subgroup of its upper hypercenter and the hypercentral residual of the group. If X is a class of groups and G is a group, we recall that the X-residual of G is the intersection G X of all normal subgroups H of G such that G/H belongs to X. If the class X is a variety of groups, then G/G X ∈ X even though the belonging fails in general. However it is worth

Supported by Proyecto MTM2010-19938-C03-03 of the Department of I+D+i of MINECO (Spain), the Department of I+D of the Government of Aragón (Spain) and FEDER funds from European Union. L. A. Kurdachenko Department of Algebra, National University of Dnepropetrovsk, 72 Gagarin Av., 49010 Dnepropetrovsk, Ukraine e-mail: [email protected] J. Otal (B) Department of Mathematics-IUMA, University of Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain e-mail: [email protected]

L. A. Kurdachenko, J. Otal

mentioning that the property holds for some classes of groups that are not varieties and some specific types of groups. For example, the class of locally nilpotent groups is not a variety, but if G is a locally finite group, then its factor-group by the locally nilpotent residual is locally nilpotent too.Also we recall that the upper central series of a group G is the series 1 = ζ0 (G) ≤ ζ1 (G) ≤ ζ2 (G) ≤ · · · ≤ ζα (G) ≤ ζα+1 (G) ≤ · · · ζγ (G), where ζ1 (G)= ζ (G) is the center of G, ζα+1 (G)/ζα (G) = ζ (G/ζα (G)) for every ordinal α, ζλ (G) = μ<λ ζμ (G) for every limit ordinal λ, and ζ (G/ζγ (G)) = 1. The term ζα (G) is said to be the αth-hypercenter of G, and the last term ζγ (G) of this series is said to be the upper hypercenter of G. The ordinal γ is said to be the central length of G and is denoted by zl(G). Suppose now that the upper central series of a group G has finite length and the factorgroup of G by its upper hypercenter is finite. Then the nilpotent residual G N of G is finite and the factor-group G/G N is nilpotent. This result is contained in the famous lectures of Hall [3, §8], and Hall called it Theorem of Baer. Actually, the result could be deduced from the results of [1], though Hall gives a different proof. Later on Otal and Peña [10] were able to extend the mentioned result. Specifically they proved that if the upper central series of G has finite length and the factor-group of G by its upper hypercenter is Chernikov, then the locally nilpotent residual R of G is Chernikov and the factor-group G/R is hypercentral (and hence R is the hypercentral residual of G). We recall that a group G is said to be a Chernikov group if it has an abelian normal subgroup D(G) that is the direct product of finitely many quasicyclic subgroups, that is D(G) = Q 1 × · · · × Q m , where Q 1 , . . . , Q m are quasicyclic, and the factor-group G/D(G) is finite (the subgroup D(G) is characteristic and is called the divisible part of G). More recently, in the paper [5], the following generalization of Baer’s theorem was obtained: if the factor-group of a group G by its upper hypercenter is finite and has order t, then the locally nilpotent residual L of G is finite and the factor-group G/L is hypercentral. Moreover, |L| ≤ tw(t), where w(t) = t m and m = 21 (log p t − 1), being p the least prime dividing t. The function w(t) appeared for the first time in the paper Wiegold [12], who specifically proved that if G is a group and |G/ζ (G)| = t, then |[G, G]| ≤ w(t). We conclude that the next natural step in this setting is the question of the study of groups whose factor-group by the upper hypercenter is Chernikov. In this paper we will focus on periodic groups with this property. For these groups we obtained a result similar to the theorem proved in [10] to which we add bounds for the numerical invariants of the locally nilpotent residual of the group in consideration. A Chernikov group G has the following family of invariants. – The index of the divisible part of G, o(G). This is simply o(G) = |G/D(G)|. – The Spectrum of G, Sp(G). This is the set of primes involved in the divisible part D(G), that is Sp(G) = (D(G)). If H is a group, we set (H ) to denote the set of primes occurring as a divisor of the order of some periodic element of H . – The minimax p-rank mm p (G) of G and the minimax rank mm(G) of G. If p is a prime and A is an abelian p-group, then the lower layer 1 (A) of A can be thought as a vector space over the prime field F p . The dimension r p (A) of this vector space is an invariant of A called the p-rank of A. In a Chernikov group G, we may write D(G) = Dr p∈Sp(G) S p ,

Groups with Chernikov factor-group by hypercentral

where S p is the Sylow p-subgroup of D(G). Actually we have S p = C1 × · · · × Cm( p) , where the direct factors C j are quasicyclic p-subgroups. It is not hard to see that the number m( p) of direct factors is exactly the p-rank of S p , which is an invariant of S p and hence of the whole group G. By definition, the number m( p) = r p (S p ) is called the minimax p-rank of G and denoted by mm p (G). Another invariant of G is the sum  m( p), p∈Sp(G) which is called the minimax rank of G and denoted by mm(G). We also recall that if A is an abelian p-group and n ≥ 1, the nth-layer of A is the subgroup n n (A) = {a ∈ A | a p = 1}. Thus the main result of this paper is the following theorem. Theorem A. Let G be a periodic group and let L be a normal subgroup of G such that G/L is a Chernikov group. If the upper hypercenter of G includes L then G has a Chernikov normal subgroup K such that G/K is hypercentral. Moreover, Sp(K ) ⊆ Sp(G/L), mm p (K ) ≤ 2mm p (G/L)(w(o(G/L))o(G/L)mm(G/L)+1 ) and o(K ) ≤ (w(o(G/L))o(G/L)mm(G/L)+1 )w(w(o(G/L))o(G/L)mm(G/L)+1 ).

2 An analogue result to Schur’s Theorem Polovickij [9] has obtained the following generalization of Schur’s theorem from finite groups to Chernikov groups: if the central factor-group G/ζ (G) of a group G is Chernikov, then the derived subgroup [G, G] of G is Chernikov too. However Polovickij’s result proves nothing on the numerical characteristic involved. Therefore we start by showing the following generalization of Polovickij’s theorem. Proposition 1 Let G be a group such that G/ζ (G) is Chernikov. Then the derived subgroup [G, G] is also Chernikov. Moreover we have o([G, G]) ≤ w(o(G/ζ (G))), mm p ([G, G]) ≤ mm p (G/ζ (G))o(G/ζ (G)) and Sp([G, G]) ⊆ Sp(G/ζ (G)). Proof In what follows we shall put mm p (G/ζ (G)) = m p,G and o(G/ζ (G)) = oG . Put Z = ζ (G) and D/Z = D(G/Z ). If d ∈ D, we define a mapping ξd : D → Z by ξd (x) = [d, x], x ∈ D. We have ξd (x y) = [d, x y] = [d, y][d, x] y = [d, y][d, x] = [d, x][d, y] = ξd (x)ξd (y), because Im(ξd ) ≤ Z = ζ (G). Thus ξd is an endomorphism of D and we have that [d, D] = Im(ξd ) ∼ = D/Ker(ξd ). Clearly Z ≤ Ker(ξd ). If Ker(ξd ) = D, then D/Ker(ξd ) is a (non-trivial) Chernikov divisible group. We have [d 2 , x] = [dd, x] = [d, x]d [d, x] = [d, x][d, x] = [d, x]2 ,

L. A. Kurdachenko, J. Otal

and, by induction, [d n , x] = [d, x]n for each n ≥ 1. Since D/Z is periodic, there exists a positive integer k such that d k ∈ Z . Then [d, x]k = [d k , x] = 1 for each x ∈ D, and we deduce that [d, D] has finite exponent and hence it cannot by divisible, a contradiction. It follows that Ker(ξd ) = D, that is C D (d) = D. Since this holds for any d ∈ D, we conclude that D is abelian. Let V = {v1 , · · · , vs } be a transversal to D in G so that s ≤ oG . Since D(G/ζ (G)) = D/ζ (G), it follows that o(G/ζ (G)) = |G : D|, and we actually have s = oG . If v ∈ V , we consider the mapping ξv : D → D given by ξv (x) = [v, x], x ∈ D. Since D is abelian, ξv is an endomorphism of D such that Im(ξv ) = [v, D] and Ker(ξv ) = C D (v), so that [v, D] = Im(ξv ) ∼ = D/Ker(ξv ) = D/C D (v). Clearly Z ≤ Ker(ξv ), so that [v, D] is Chernikov divisible, mm p ([v, D]) ≤ m p,G and Sp([v, D]) ⊆ Sp(G/Z ). We have [G, D] = [v1 , D] · · · [vs , D] (see [4, Lemma 1.1], for example). It follows that [G, D] is divisible Chernikov, mm p ([G, D]) ≤ s · m p,G ≤ oG · m p,G and Sp([G, D]) ⊆ Sp(G/Z ). The center of G/[G, D] includes D/[G, D] and so (G/[G, D])/ζ (G/[G, D]) is finite and has order at most oG . Then [G/[G, D], G/[G, D]] is finite and has order at most w(oG ) = otG , where t = 21 (log2 oG − 1) [12]. Since [G/[G, D], G/[G, D]] = [G, G]/[G, D], [G, G] is Chernikov, mm p ([G, G]) ≤ oG m p,G , o([G, G]) ≤ w(oG ) and Sp([G, G]) ⊆ Sp(G/Z ).  Corollary 1 Let G be a group and L be a subgroup of ζ (G). If G/L is Chernikov, then the derived subgroup [G, G] is also Chernikov. Moreover, o([G, G]) ≤ w(o(G/L)), mm p ([G, G]) ≤ mm p (G/L)o(G/L) and Sp([G, G]) ⊆ Sp(G/L). Proof G/ζ (G)) is Chernikov, o(G/ζ (G)) ≤ o(G/L), mm p (G/ζ (G)) ≤ mm p (G/L) and Sp(G/ζ (G)) ⊆ Sp(G/L). 

3 Direct decompositions in modules For the sequel we need the following module-theoretical concepts. Let G be a group, R a ring and A an RG-module. Then the set ζ RG (A) = {a ∈ A | a(g − 1) = 0 for each element g ∈ G} is a submodule called the RG-center of A. The upper RG-central series of A is, {0} = A0 ≤ A1 ≤ · · · ≤ Aα ≤ Aα+1 ≤ · · · Aγ , where A1 = ζ RG (A), Aα+1 /Aα = ζ RG (A/Aα ), α < γ , Aλ = ∪μ<λ Aμ for a limit ordinal λ and ζ RG (A/Aγ ) = {0}. The last term Aγ of this series is called the upper RG-hypercenter ∞ (A). The RG-module A is said to be RG-hypercentral if of A and will be denoted by ζ RG A = Aγ happens and RG-nilpotent if γ is finite. If B and C are RG-submodules of A and B ≤ C, then the factor C/B is said to be G-central if G = C G (C/B) and G-eccentric otherwise. An RG-submodule C of A is said

Groups with Chernikov factor-group by hypercentral

to be RG-hypereccentric if C has an ascending series of RG-submodules {0} = C0 ≤ C1 ≤ · · · ≤ Cα ≤ Cα+1 ≤ · · · Cγ = C whose factors Cα+1 /Cα are G-eccentric simple RG-modules. After Zaitsev [13], an RG-module A is said to have the Z -RG-decomposition (the Z decomposition if R = Z) if one has ∞ A = ζ RG (A) ⊕ E ∞ RG (A),

where E ∞ RG (A) is the unique maximal RG-hypereccentric RG-submodule of A. We note that a given maximal E includes every RG-hypereccentric RG-submodule B and, in particular, it is unique. For, if (B + E)/E is non-zero, it has to include a non-zero simple RG-submodule U/E. Since (B + E)/E ∼ = B/(B ∩ E), U/E is RG-isomorphic to some simple RG-factor ∞ (A), of B and it follows that C G (U/E) = G. On the other hand, (B + E)/E ≤ A/E ≤ ζ RG that is G/C G (U/E) = G. This contradiction shows that B ≤ E, as claimed. Lemma 1 Let G be a finite nilpotent group and 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 of finite ZG-submodules. By the results of [13], every member of L has the Z -decomposition. Let B, C ∈ L such that B ≤ C. We have the decompositions B = ζZ∞G (B) ⊕ E Z∞G (B) and C = ζZ∞G (C) ⊕ E Z∞G (C). Clearly, ζZ∞G (B) ≤ ζZ∞G (C). As we remarked above, E Z∞G (C) includes every ZGhypereccentric ZG-submodule, in particular, E Z∞G (B) ≤ E Z∞G (C). It follows that   ζZ∞G (A) = ζZ∞G (B) and E Z∞G (A) = E Z∞G (B) B∈L

B∈L

and we clearly have A = ζZ∞G (A)



E Z∞G (A), 

as required.

Lemma 2 Let G be a hypercentral Chernikov group and A be a ZG-module. Suppose that A satisfies the two following conditions: (i) the additive group of A is periodic; and (ii) the additive group of A/ζZ∞G (A) is Chernikov. Then A has the Z -decomposition. Proof Suppose first that the additive group of A is a p-group for some prime p. Put Z = ζZ∞G (A). Since the additive group of A/Z is Chernikov, the ZG-module A/Z is artinian. Then A/Z has the Z -decomposition [13], that is A/Z = Z 1 /Z ⊕ E/Z , where Z 1 /Z = ζZ∞G (A/Z ) and E/Z = E Z∞G (A/Z ). Clearly the ZG-submodule Z 1 is ZGhypercentral and therefore Z 1 ≤ ζZ∞G (A) = Z . Thus A/Z = E Z∞G (A/Z ), that is A/Z is

L. A. Kurdachenko, J. Otal

ZG-hypereccentric. Since the additive group of A/Z is Chernikov, A has an ascending series of ZG-submodules  Bn = A, {0} = B0 ≤ B1 = Z ≤ B2 ≤ · · · ≤ Bn ≤ · · · n≥1

where Bn+1 /Bn = 1 (A/Bn ) for every n ≥ 1. Since A/B1 is ZG-hypereccentric, B2 /B1 has a finite series of ZG-submodules B1 = B1,0 ≤ B1,1 ≤ · · · ≤ B1,k = B2 , whose factors B1, j+1 /B1, j are G-eccentric simple ZG-modules. Since G is hypercentral, each G/C G (B1, j+1 /B1, j ) is a p  -group (see [6, Theorem 3.1] for example). Note that the finiteness of B1, j+1 /B1, j implies that G/C G (B1, j+1 /B1, j ) is a finite p  -group. Since G is a periodic hypercentral group, we have the direct decomposition G = P × Q, where P is a Sylow p-subgroup of G and Q is a p  -subgroup of G. Therefore P ≤ C G (B1, j+1 /B1, j ) and then we have C G (B1, j+1 /B1, j ) = P × C Q (B1, j+1 /B1, j ) for every 0 ≤ j ≤ k − 1. Let Q 1 = C Q (B1,1 /B1,0 ) ∩ · · · ∩ C Q (B1,k /B1,k−1 ) so that the index |Q : Q 1 | is finite. If x ∈ Q 1 , then x acts trivially on every factor of the series B1 = B1,0 ≤ B1,1 ≤ · · · ≤ B1,k = B2 p  -element,

and, being a x ∈ C Q (B2 /B1 ). In other words, Q 1 = C Q (B2 /B1 ). Consider now ξ : a → a p , a ∈ B3 /B1 . Clearly ξ is a ZG-endomorphism of A such that Ker(ξ ) = 1 (B3 /B1 ) = B2 /B1 and Im(ξ ) ≤ 1 (B3 /B1 ) = B2 /B1 . Since (B3 /B2 ) ∼ = ZG (B3 /B1 )/1 (B3 /B1 ) = (B3 /B1 )/Ker(ξ ) ∼ = ZG Im(ξ ) ≤ 1 (B3 /B1 ) = B2 /B1 , C Q (B3 /B2 ) ≥ C Q (B2 /B1 ) = Q 1 . Every x ∈ Q 1 acts trivially on B3 /B2 and B2 /B1 . Since x is a p  -element, x ∈ C Q (B3 /B1 ). In other words, Q 1 = C Q (B3 /B1 ). Proceeding in this way and applying induction, we obtain that Q 1 = C Q (Bn /B1 ) for every n > 1. It follows that Q 1 = C Q (A/B1 ). Since B1 is ZG-hypercentral and Q 1 is a p  -subgroup, Q 1 = C Q (A). Therefore Q/C Q (A) is finite, and, without loss of generality, we can assume that Q is finite. By Lemma 1, A has the Z -Z Q-decomposition, that is A = ζZ∞Q (A) ⊕ E 1 , where E 1 = E Z∞Q (A) is the maximal Z Q-hypereccentric Z Q-submodule of A. Since A/Z has an ascending series of ZG-submodules whose factors are Q-eccentric ZG-simple, Z = ζZ∞Q (A) and E 1 ∼ = Z Q A/Z . Pick x ∈ G and let U ≤ V be Z Q-submodules of E 1 such that V /U is a simple Z Q-submodule that also is Q-eccentric. Then Q = C Q (V /U ) and therefore there are elements v ∈ V and y ∈ Q such that vy ∈ / v + U . Since Q is normal in G, y1 = x −1 yx ∈ Q. Suppose that (vx)y1 ∈ vx + U x, that is (vx)y1 = vx + ux = (v + u)x for some u ∈ U. We have (vx)y1 = v(x y1 ) and x y1 = x x −1 yx = yx and so (vx)y1 = v(x y1 ) = v(yx) = (vy)x. Then (vy)x = (vx)y1 = (v + u)x, which implies that vy = v + u ∈ v + U , a

Groups with Chernikov factor-group by hypercentral

contradiction. This contradiction shows that (vx)y1 ∈ / vx +U x and hence the factor V x/U x is Q-eccentric. It follows that E 1 x is a Q-hypereccentric Z Q-submodule for every x ∈ G. Since E Z∞Q (A) includes every Q-hypereccentric Z Q-submodule, E 1 x ≤ E 1 . This holds for each x ∈ G, therefore E 1 is in fact a ZG-submodule. Then E 1 ∼ = ZG A/Z , so that E 1 is G-hypereccentric and E 1 = E Z∞G (A). In the general case we put π = (A/Z ) so that the set π is finite since A/Z is Chernikov. We have the decomposition  A= A p, p∈(A)

where A p is the p-component of A. Given p ∈ π, by the result proved in the above paragraphs, we have A p = ζZ∞G (A p ) ⊕ E Z∞G (A p ). If S is the π-component of A, then we have   Ap = (ζZ∞G (A p ) ⊕ E Z∞G (A p )) S= p∈π

p∈π

  =( ζZ∞G (A p )) ⊕ ( E Z∞G (A p )) = ζZ∞G (S) ⊕ E ∞ p∈π (S) p∈π

p∈π

and so S has the Z -decomposition. If R is the π  -component of A, since (R) ∩ (A/Z ) =, R ≤ Z . Then A = S ⊕ R = ζZ∞G (S) ⊕ E Z∞G (S) ⊕ R = ζZ∞G (A) ⊕ E Z∞G (S) = ζZ∞G (A) ⊕ E Z∞G (A) and therefore A has the Z -decomposition, as required.



4 More auxiliary results Lemma 3 Let G be a periodic group and L be a normal subgroup of G such that G/L is a Chernikov group. If the upper hypercenter of G includes L, then G/C G (L) is hypercentral. Proof Clearly G is FC-hypercentral (see [11, §4] for example). Since an FC-hypercentral locally nilpotent group is hypercentral (see [11, Corollary 1 to Theorem 4.38] for example), it will be suffice to show that G/C G (L) is locally nilpotent. Put C = C G (L) and let F/C be an arbitrary finite subgroup of G/C. Then there exists a finite subgroup K such that F = K C. Let L be the family of all finite subgroups of L. Since K is finite, U K is also finite for each subgroup U ∈ L. Clearly the upper hypercenter of U K K includes U K , and therefore, by Kaluzhnin’s theorem [8], we obtain that K /C K (U K ) is nilpotent. Let ρ = {|C K (U K )| | U ∈ L}. Pick V ∈ L such that |C K (V K )| is the least number of ρ. Given U ∈ L, we set W = U, V . Then W ∈ L and W K = U K V K . Therefore C K (W K ) ≤ C K (V K ) and then |C K (W K )| ≤ |C K (V K )|. By the choice of V , we see that C K (W K ) = C K (V K ). Let L1 = {U, V  K | U ∈ L}

L. A. Kurdachenko, J. Otal

so that L1 is clearly another local system of L that satisfies C K (W K ) = C K (V K ) for every subgroup W K ∈ L1 . It follows that C K (L) = C K (V K ) and in particular K /C K (L) is nilpotent. We have F/C = K C/C ∼ = K /(K ∩ C) = K /C K (L), which shows that F/C is nilpotent. Hence G/C G (L) is locally nilpotent.



An abelian normal subgroup A of a group G is said to be G-quasifinite if A is infinite but every proper G-invariant subgroup of A is finite. It is not hard to see that if A is not G-simple then A is a p-group for some prime p. Moreover, if A is Chernikov then A has to be divisible and is the union of its G-invariant finite subgroups. Lemma 4 Let G be a Chernikov group. Suppose that P is the Sylow p-subgroup of D(G) for some prime p. Then P includes a finite G-invariant subgroup F such that P/F = Q 1 /F × · · · × Q m /F, where Q 1 /F, . . . , Q m /F are G-quasifinite G-invariant subgroups. Moreover, if |G/C G (P)| = p k q, where p and q are coprime, then |F| ≤ p kr p (P) . In particular, if p does not divide G/C G (P), then F = 1. Proof Let Q = {Q | Q is an infinite G-invariant subgroup of P}.

Since P is Chernikov, it satisfies the minimal condition on subgroups, and therefore Q has a minimal element, say Q 1 . Clearly Q 1 is divisible and G-quasifinite. It turns out that Q 1 has a direct complement in P ([2, Theorem 21.2]) and then P includes a G-invariant subgroup R1 such that – P = Q 1 R1 , – Q 1 ∩ R1 = S is finite, and – S t = 1, where t = |G/C G (P)|. (see [7, Corollary 5.11] for example). In particular, S ≤ t (P) and then |S| ≤ p kr p (Q 1 ) . We have P/S = Q 1 /S × R1 /S. Proceeding in the same way with the subgroup R1 , after finitely many steps we reach the required result.  Corollary 2 Let G be a Chernikov group and put D = D(G). Then D includes a finite G-invariant subgroup F such that D/F = Q 1 /F × · · · × Q m /F, where Q 1 /F, . . . , Q m /F are G-quasifinite G-invariant subgroups. Moreover, |F| ≤ |G/D|mm(G) . Proof Suppose that |G/D| = t = p1k1 · · · psks is the decomposition of t as powers of distinct primes and put π = { p1 , · · · , ps }. We have D = P1 × · · · × Ps × R,

Groups with Chernikov factor-group by hypercentral

where P j is the Sylow p j -subgroup of D for 1 ≤ j ≤ s and R is the Hall π  -subgroup of D. Put m( p) = r p (D) for every prime p and let m = m( p1 ) + · · · + m( ps ). Applying Lemma 4 to every Sylow p-subgroup of D, we obtain the existence of a finite G-invariant subgroup F such that D/F = Q 1 /F × · · · × Q m /F, where Q 1 /F, . . . , Q m /F are G-quasifinite G-invariant subgroups and moreover k m( p1 )

|F| ≤ p11

k m( ps )

· · · ps s

≤ p1 k1 m · · · ps ks m = ( p1 k1 m · · · ps ks )m = t m ≤ t mm(G)

as required. Let G be a group and H be a normal subgroup of G. It is said that H is Gnilpotent (respectively, G-hypercentral) if H has a finite series (respectively, an ascending series) of G-invariant subgroups, whose factors are G-central.  Lemma 5 Let G be a group and let D be a divisible abelian normal p-subgroup of G, where p is a prime. If there exists a positive integer n such that the factor n+1 (D)/n (D) is G-hypercentral, then D is G-hypercentral. Proof Given positive integers m < j, set k = p m , and consider the mapping ξ :  j (D) →  j−m (D) given by ξ(d) = d k for each element d ∈  j (D). Clearly this mapping is a G-homomorphism. Since D is divisible, actually ξ is an epimorphism. Thus we have  j (D)/m (D) =  j (D)/Ker(ξ ) ∼ = G Im(ξ ) =  j−m (D). In particular, n+1 (D)/n (D) is G-isomorphic to the factors 1 (D), 2 (D)/1 (D), . . . , n (D)/n−1 (D). Since n+1 (D)/n (D) is also G-isomorphic to the factors  j+n+1 (D)/  j+n (D) for every j ≥ 1, it follows that D has an ascending series of G-invariant subgroups, whose factors are G-central, as required.  Lemma 6 Let G be a Chernikov group and R be the locally nilpotent residual of G. If Q is a G-quasifinite G-invariant subgroup of D(G), then either Q ≤ R or Q is G-hypercentral. Proof Note that Q is a p-group for some prime p. Put D = D(G). If R does not include Q, then Q ∩ R is a proper G-invariant subgroup of Q. Since Q is G-quasifinite, Q ∩ R is finite. Then there exists a positive integer s such that Q ∩ R ≤ s (Q). Since the factor-group G/R is locally nilpotent, the factor Q R/R is G-hypercentral. Also Q R/R ∼ = G Q/(Q ∩ R), so that Q/(Q ∩ R) is clearly G-hypercentral. It follows that the factor s+1 (Q)/s (Q) is G-nilpotent. By Lemma 5, Q is G-hypercentral.  Lemma 7 Let G be a Chernikov group. If R is the locally nilpotent residual of G, then o(R) ≤ w(o(G))o(G)mm(G)+1 . Proof Put D = D(G). Suppose first that D = Q1 × · · · × Qm , where Q 1 , . . . , Q m are G-quasifinite G-invariant subgroups. If R does not include some Q j , then, by Lemma 6, Q j is G-hypercentral. Therefore we suppose that R includes Q 1 , . . . , Q k , but it does not include Q k+1 , . . . , Q m . Put U = Q 1 × · · · × Q k and V = Q k+1 × · · · × Q m . Then U ≤ R and D/U is G-hypercentral. It follows that the upper hypercenter of G/U has finite index that is bounded by |G/D| = o(G). By [5, Theorem B], the locally nilpotent

L. A. Kurdachenko, J. Otal

residual R1 /U of G/U is finite and has order at most w(o(G))o(G). Clearly R ≤ R1 and then o(R) ≤ w(o(G))o(G). In the general case, by Corollary 2, D includes a finite G-invariant subgroup E such that D/E = S1 /E × · · · × S /E, where S1 /E, . . . , S /E are G-quasifinite G-invariant subgroups, and we have that |E| ≤ |G/D|mm(G) = o(G)mm(G) . If R2 /E is the locally nilpotent residual of G/E, then we apply the result of the above paragraph to deduce that o(R2 /E) ≤ w(o(G))o(G). Clearly R ≤ R2 and then o(R) ≤ w(o(G))o(G)o(G)mm(G) = w(o(G))o(G)mm(G)+1 , 

as required.

5 Proof of Theorem A Let R1 /L be the locally nilpotent residual of G/L. By Lemma 7, o(R1 /L) ≤ w(o(G/L))o(G/L)mm(G/L)+1 . Let R be the locally nilpotent residual of G. By Lemma 3, G/C G (L) is a hypercentral group and hence R ≤ C G (L). If u ∈ R and z ∈ L, then [u, z] = 1. It follows that L ≤ C G (R), and in particular G/C G (R) is a Chernikov group. Clearly R ∩ L ≤ ζ (R). Moreover, by Lemma 7, o(R/(L ∩ R)) = o(R L/L) ≤ w(o(G/L))o(G/L)mm(G/L)+1 . By Corollary 1, the derived subgroup D = [R, R] is Chernikov and moreover o(D) ≤ w(o(R L/L)) ≤ w(w(o(G/L))o(G/L)mm(G/L)+1 ), Sp(D) ⊆ Sp(R L/L) ⊆ Sp(G/L), and mm p (D) ≤ mm p (G/L)(w(o(G/L))o(G/L)mm(G/L)+1 ). The subgroup D is G-invariant and R/D is abelian. Then R/D ≤ C G/D (R/D). The latter implies that (G/D)/C G/D (R/D) is a hypercentral group. Since L ≤ C G (R), L D/D ≤ C G/D (R/D) and hence (G/D)/C G/D (R/D) is Chernikov. By Lemma 2, R/D has the Z decomposition, that is R/D = ζZ∞G (R/D) ⊕ E Z∞G (R/D). Since (L ∩ R)D/D ≤ ζZ∞G (R/D), E Z∞G (R/D) := K /D is a Chernikov subgroup. As R/K = ζZ∞G (R/K ), G/K is hypercentral since G/R is hypercentral. Furthermore, Sp(K ) ⊆ Sp(D) ∪ Sp(R/(L ∩ R)) ⊆ Sp(G/L), mm p (K ) ≤ mm p (D) + mm p (R/(L ∩ R)) ≤ 2mm p (G/L)(w(o(G/L))o(G/L)mm(G/L)+1 )

Groups with Chernikov factor-group by hypercentral

and o(K ) ≤ o(R/(R ∩ L))o(D) ≤ (w(o(G/L))o(G/L)mm(G/L)+1 )w(w(o(G/L))o(G/L)mm(G/L)+1 ), as required.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

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