On modules over group rings of locally soluble groups for a ring of p -adic integers

On modules over group rings of locally soluble groups for a ring of p -adic integers

The author studies the Zp∞G-module A such that Zp∞ is a ring of p-adic integers, a group G is locally soluble, the quotient module A/CA(G) is not Artinian Zp∞-module, and the system of all subgroups H≤G for which the quotient modules A/CA(H) are not Artinian Zp∞-modules satisfies the minimal conditi...

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Datum:2009
1. Verfasser: Dashkova, O.Yu.
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Veröffentlicht: Інститут прикладної математики і механіки НАН України 2009
Schriftenreihe:Algebra and Discrete Mathematics
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Zitieren:On modules over group rings of locally soluble groups for a ring of p -adic integers / O. Yu. Dashkova // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 32–43. — Бібліогр.: 9 назв. — англ.

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spelling irk-123456789-1545732019-06-16T01:25:29Z On modules over group rings of locally soluble groups for a ring of p -adic integers Dashkova, O.Yu. The author studies the Zp∞G-module A such that Zp∞ is a ring of p-adic integers, a group G is locally soluble, the quotient module A/CA(G) is not Artinian Zp∞-module, and the system of all subgroups H≤G for which the quotient modules A/CA(H) are not Artinian Zp∞-modules satisfies the minimal condition on subgroups. It is proved that the group G under consideration is soluble and some its properties are obtained. 2009 Article On modules over group rings of locally soluble groups for a ring of p -adic integers / O. Yu. Dashkova // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 32–43. — Бібліогр.: 9 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20F19; 20H25. http://dspace.nbuv.gov.ua/handle/123456789/154573 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The author studies the Zp∞G-module A such that Zp∞ is a ring of p-adic integers, a group G is locally soluble, the quotient module A/CA(G) is not Artinian Zp∞-module, and the system of all subgroups H≤G for which the quotient modules A/CA(H) are not Artinian Zp∞-modules satisfies the minimal condition on subgroups. It is proved that the group G under consideration is soluble and some its properties are obtained.
format Article
author Dashkova, O.Yu.
spellingShingle Dashkova, O.Yu.
On modules over group rings of locally soluble groups for a ring of p -adic integers
Algebra and Discrete Mathematics
author_facet Dashkova, O.Yu.
author_sort Dashkova, O.Yu.
title On modules over group rings of locally soluble groups for a ring of p -adic integers
title_short On modules over group rings of locally soluble groups for a ring of p -adic integers
title_full On modules over group rings of locally soluble groups for a ring of p -adic integers
title_fullStr On modules over group rings of locally soluble groups for a ring of p -adic integers
title_full_unstemmed On modules over group rings of locally soluble groups for a ring of p -adic integers
title_sort on modules over group rings of locally soluble groups for a ring of p -adic integers
publisher Інститут прикладної математики і механіки НАН України
publishDate 2009
url http://dspace.nbuv.gov.ua/handle/123456789/154573
citation_txt On modules over group rings of locally soluble groups for a ring of p -adic integers / O. Yu. Dashkova // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 32–43. — Бібліогр.: 9 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT dashkovaoyu onmodulesovergroupringsoflocallysolublegroupsforaringofpadicintegers
first_indexed 2025-07-14T04:40:56Z
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fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 1. (2009). pp. 32 – 43 c© Journal “Algebra and Discrete Mathematics” On modules over group rings of locally soluble groups for a ring of p-adic integers O. Yu. Dashkova Communicated by L. A. Kurdachenko Abstract. The author studies the Zp∞G-module A such that Zp∞ is a ring of p-adic integers, a group G is locally solu- ble, the quotient module A/CA(G) is not Artinian Zp∞ -module, and the system of all subgroups H ≤ G for which the quotient modules A/CA(H) are not Artinian Zp∞-modules satisfies the min- imal condition on subgroups. It is proved that the group G under consideration is soluble and some its properties are obtained. 1. Introduction At present a number of works of scientists is devoted to the investigation of infinite dimensional linear groups with the large system of sugroups which are similar to finite dimensional groups. The one of basic notions which is applied here is the central dimension of subgroup, i.e. the codi- mension of centralizer of this subgroup in the vector space. The linear groups of finite central dimension turned out very similar to the usual finite dimensional linear groups. The one of directions in the theory of infinite dimensional linear groups is engaged in the study of linear groups with some essential restrictions on the family of subgroups of infinite cen- tral dimension. The natural extension of the theory of linear groups is the theory of modules over group rings. In this work the one of ana- logues of the notion of central dimension is considered. In the theory of modules there exists a number of generalizations of finite dimensional vector space. These are modules with finite composition series, finitely generated modules, Noetherian modules, Artinian modules. 2000 Mathematics Subject Classification: 20F19; 20H25. Key words and phrases: Linear group, Artinian module, locally soluble group. Jo u rn al A lg eb ra D is cr et e M at h .O. Yu. Dashkova 33 The broad class of modules over group rings is Artinian modules over group rings. Remind that a module is called Artinian if partially ordered set of all submodules of this module satisfies the minimal condition. It should be noted that many problems of Algebra require the investigation of some specific Artinian modules. The Artinian modules over group rings with different restrictions on groups were studied in [1]. Naturally, it is arised the question on investigation of modules over group rings which are not Artinian but which are similar to Artinian modules in some sense. Let A be a DG-module where D is Dedekind domain, G is a group. If H ≤ G then the quotient module A/CA(H) is called the cocentralizer of H in module A. The subject of investigation of this paper is a DG- module A where D = Zp∞ is a ring of p-adic integers. Let A be Zp∞G-module such that the cocentralizer of the group G in module A is not Artinian Zp∞-module and let Lnad(G) be a system of all subgroups of G such that its cocentralizers in module A are not Artinian Zp∞-modules. The system of Lnad(G) is the partially ordered set relative to usual inclusion of subgroups. If the system Lnad(G) satisfies the minimal condition on subgroups then we shall say that the group G satisfies the condition min − nad. Later on in the work it is considered Zp∞G-module A such that CG(A) = 1. Let A be a Zp∞G-module such that the cocentralizer of group G in module A is not Artinian Zp∞-module and the group G is locally soluble and satisfies the condition min − nad. In the work it is proved that in this case the group G is soluble, and some its properties are obtained. The basic results of the work are the following theorems 1.1, 1.2 and 1.3. Theorem 1.1. Let A be Zp∞G-module, G be a locally soluble group which satisfies the condition min − nad. If the cocentralizer of group G in module A is not Artinian Zp∞-module then the group G is soluble. Theorem 1.2. Let A be Zp∞G-module, G be a locally soluble group which satisfies the condition min − nad. If the cocentralizer of group G in module A is not Artinian Zp∞-module then the group G contains the normal nilpotent subgroup H such that the quotient group G/H is Chernikov group. Theorem 1.3. Let A be Zp∞G-module, G be a locally soluble group. Suppose that the cocentralizer of group G in module A is not Artinian Zp∞-module. If the cocentralizer of each proper subgroup of group G in module A is Artinian Zp∞-module then G ≃ Cq∞ for some prime q. Jo u rn al A lg eb ra D is cr et e M at h .34 On modules over group rings of locally soluble groups 2. Preliminary results We mention some elementary facts on Zp∞G-modules. Recall that if K ≤ H ≤ G and the cocentralizer of subgroup H in module A is Artinian Zp∞- module then the cocentralizer of subgroup K in module A is Artinian Zp∞-module also. If U, V ≤ G such that their cocentralizers in module A are Artinian Zp∞-modules then the quotient module A/(CA(U)∩CA(V )) is Artinian Zp∞-module also. Therefore the cocentralizer of subgroup < U, V > in module A is Artinian Zp∞-module. Suppose now that the group G satisfies the condition min − nad. If H1 > H2 > H3 > ... is an infinite strictly descending chain of subgroups of G, then there is a natural number n such that the the cocentralizer of subgroup Hn in module A is Artinian Zp∞-module. Moreover, if N is a normal subgroup of G and the cocentralizer of subgroup N in module A is not Artinian Zp∞-module then the quotient group G/N satisfies the minimal condition on subgroups. Lemma 2.1 [2]. Let A be Zp∞G-module and suppose that G satisfies the condition min− nad. Let X, H be subgroups of G and Λ be an index set such that: (i) X = Drλ∈ΛXλ, where 1 6= Xλ is an H-invariant subgroup of X, for each λ ∈ Λ. (ii) H ∩ X ≤ Drλ∈ΓXλ for some subset Γ of Λ. If Ω = Λ\Γ is infinite, then the cocentralizer of subgroup H in module A is Artinian Zp∞-module. Lemma 2.2. Let A be Zp∞G-module and suppose that G satisfies the condition min − nad. Let H, K be subgroups of G such that K is a normal subgroup of H and suppose that there exists an index set Λ and subgroups Hλ of G such that K ≤ Hλ for all λ ∈ Λ. Suppose that H/K = Drλ∈ΛHλ/K and that Λ is infinite. Then the cocentralizer of subgroup H in module A is Artinian Zp∞-module. Proof. Suppose that Λ is infinite. Let Γ and Ω be infinite disjoint sub- sets of Λ such that Λ = Γ ∪ Ω. Let U/K = Drλ∈ΓHλ/K, let V/K = Drλ∈ΩHλ/K, and let Γ1 ⊇ Γ2 ⊇ ... be a strictly descending chain of subsets of Γ. Then we obtain an infinite descending chain < U, Hλ|λ ∈ Γ1 > >< U, Hλ|λ ∈ Γ2 > > ... . of subgroups. It follows from the condition min−nad that the cocentra- lizer of subgroup U in module A is Artinian Zp∞-module. Likewise, the cocentralizer of subgroup V in module A is Artinian Zp∞-module. Since H = UV , it follows that the cocentralizer of subgroup H in module A is Artinian Zp∞-module. The lemma is proved. Jo u rn al A lg eb ra D is cr et e M at h .O. Yu. Dashkova 35 Lemma 2.3. Let A be Zp∞G-module and suppose that G satisfies the condition min − nad. If the element g ∈ G has infinite order, then the cocentralizer of subgroup < g > in module A is Artinian Zp∞-module. Proof. Let p, q be distinct primes greater than 3 and let u = gp, v = gq. Then there is an infinite descending chain < u >>< u2 >>< u4 >> . . . Therefore for some natural number k, the cocentralizer of subgroup < u2k > in module A is Artinian Zp∞-module. Simirlarly, there exists a natural number l such that the cocentralizer of subgroup < v3l > in module A is Artinian Zp∞-module also. Therefore the cocentralizer of subgroup < g >=< u2k >< v3l > in module A is Artinian Zp∞-module. The lemma is proved. The following result gives an important information about the derived quotient group. Lemma 2.4. Let A be Zp∞G-module. Suppose that G satisfies the con- dition min− nad and the cocentralizer of G in module A is not Artinian Zp∞-module. Then the quotient group G/G′ is Chernikov group. Proof. Suppose for a contradiction that G/G′ is not Chernikov group. Let S be a set of all subgroups H ≤ G such that H/H ′ is not Chernikov group and the cocentralizer of subgroup H in module A is not Artinian Zp∞-module. Since G ∈ S then S 6= ∅. Since S satisfies the minimal condition it has a minimal element D. If U , V are proper subgroups of D such that D = UV and U ∩ V = D′, then at least for one of these subgroups, U say, the cocentralizer of U in module A is not Artinian Zp∞-module. The choice of D implies that U/U ′ is Chernikov group. Hence, U/D′ ≃ (U/U ′)/(D′/U ′) is Chernikov group too. Since the co- centralizer of subgroup U in module A is not Artinian Zp∞-module, it follows that the abelian group D/U is also Chernikov group. Hence D/D′ is Chernikov group. Contrary to the choice of D. Therefore the quotient group D/D′ is indecomposable. Hence D/D′ is isomorphic to the sub- group of Cq∞ for some prime q. Contradiction. The lemma is proved. Let A be Zp∞G-module and G satisfies the condition min−nad. Let AD(G) be the set of all elements x ∈ G such that the cocentralizer of group < x > in module A is Artinian Zp∞-module. Since CA(xg) = CA(x)g for all x, g ∈ G then AD(G) is a normal subgroup of G. Lemma 2.5. Let A be Zp∞G-module. Suppose that G satisfies the con- dition min− nad and the cocentralizer of G in module A is not Artinian Zp∞-module. Then G either is periodic or G = AD(G). Jo u rn al A lg eb ra D is cr et e M at h .36 On modules over group rings of locally soluble groups Proof. We suppose to the contrary that G is neither periodic nor G 6= AD(G). Let S be the set of all subgroups H ≤ G such that H is non- periodic group and H 6= AD(H). Then S is non-empty. If H 6= AD(H), then there is an element h ∈ H such that the quotient module A/CA(h) is not Artinian Zp∞-module. Hence S ⊆ Lnad(G) and S therefore has the minimal condition. Let D be a minimal element of S, let L = AD(D). Note that L 6= 1, since D is non-periodic. Let L ≤ W ≤ D and W 6= D. By lemma 2.3 L contains all elements of infinite order of non-periodic subgroup D then W is non-periodic subgroup. Therefore W = AD(W ) so W ≤ L. Hence D/L has order q for some prime q. Let x ∈ D \L. If a is an element of infinite order, then the minimal choice of D implies that < x, a >= D. Since |D : L| is finite and D is a finitely generated subgroup then by theorem 1.41 [3] L is also finitely generated subgroup. Since L = AD(L), the quotient module A/CA(L) is Artinian Zp∞-module. Since L is a normal subgroup of D, then C = CA(L) is Zp∞G-submodule of the module A. If R = CD(A/C), then R is a normal subgroup of the group D. Since the quotient module A/C is Artinian Zp∞-module, and by the theorem 7.13 [4] A/C = A1/C ⊕ A2/C ⊕ ... ⊕ An/C, i = 1, 2, ...n, where each direct summand is either Prüfer Zp∞-module or finitely gen- erated Zp∞-module. In the case D = Zp∞ for maximal ideal P of D the additive group of D/P has the order p. By corollary 1.28 [4] D/P k and P/P k+1 are isomorphic as D-modules for any k = 1, 2, ..., n, ... . In particular, the additive group of D/P k is a cyclic group of order pk. Let D/P k =< ak >. We can define the mapping πk+1 k : D/P k −→ D/P k+1 such that πk+1 k (ak) = pak+1. Therefore we can consider the in- jective limit of the family of D-modules D/P k, k = 1, 2, ..., n, ... . From the choice of a1 it followes that pa1 = 0. By the definition of Prüfer bfZp∞-module (see chapter 5 [4]) this module is the injective limit of the family of D-modules D/P k, k = 1, 2, ..., n, ... . In the case D = Zp∞ the additive group of Prüfer Zp∞-module is quasicyclic p-group. Since each ideal of Zp∞ has in this ring finite index (see chapter 8 [5]), then a finitely generated Zp∞-module is finite. Therefore the additive group of the quotient module A/C is Chernikov group, and its divisible part is p-group. By theorem 60.1.1 [6] the quotient group D/R is isomorphic to some subgroup of GL(r,Zp∞). Let U be a normal subgroup of D of finite index. Then U is not periodic and so < U, x > is not periodic and < U, x >6= AD(< U, x >). The minimal choice of D implies that D =< U, x >. Since U is a normal subgroup of D then the quotient group D/U is a cyclic group. Therefore D/U is an abelian quotient group. If E is the finite residual of D, it follows that D/E is abelian. Since E ≤ R, then the quotient group D/R is also abelian. Therefore D/(R ∩ L) is Jo u rn al A lg eb ra D is cr et e M at h .O. Yu. Dashkova 37 abelian. R ∩ L is a subgroup of stabilizer of the series of length 2, and therefore it is abelian. So that D is a finitely generated metabelian group. By a theorem of P.Hall (theorem 9.51 [3]) D is residually finite group. As above, D is therefore abelian. Since D = U < x > for every sub- group U of finite index, it follows that D is infinite cyclic. By lemma 2.3 D = AD(D). The contradiction. The lemma is proved. 3. Locally soluble groups with min-nad Let G be a locally soluble group and the quotient module A/CA(G) be Artinian Zp∞-module. As in the proof of lemma 2.5, the quotient group G/CG(A/CA(G)) is isomorphic to the locally soluble subgroup of GL(r,Zp∞). Since Zp∞ is an integral ring then it can be imbeded in the field F . Therefore the quotient group G/CG(A/CA(G)) is isomorphic to some locally soluble subgroup of the linear group GL(r, F ). Hence by corollary 3.8 [7] the quotient group G/CG(A/CA(G)) is soluble. Since CG(A/CA(G)) is abelian group then G is a soluble group. Therefore it is necessary to concentrate attenton on the study of locally soluble groups G with min− nad for which the quotient module A/CA(G) is not Artinian Zp∞-module. Lemma 3.1. Let A be Zp∞G-module, G is a periodic locally soluble group which satisfies the condition min − nad, and the cocentralizer of G in module A is not Artinian Zp∞-module. Then G either satisfies the minimal condition on subgroups or G = AD(G). Proof. We suppose to the contrary that G neither satisfies the minimal condition for subgroups nor G 6= AD(G). Let S be a set of subgroups H ≤ G such that H does not satisfies the minimal condition for subgroups and not H 6= AD(H). Then S 6= ∅ and satisfies the minimal condition. Let D be a minimal element of S and L = AD(D). There is an infinite strictly descending chain of subgroups of D: H1 > H2 > H3 > .... Since D has min−nad, there is a natural number d such that the cocen- tralizer of subgroup Hd in module A is Artinian Zp∞-module. Clearly Hd ≤ L and hence L does not satisfy the minimal condition. It fol- lows that if x ∈ D \ L, then < x, L >= D, by the minimal choice of D. Therefore D/L has prime order q, for some prime q. Replacing x by a suitable power if necessary, we may assume that x has order qr for some natural number r. Since D is not Chernikov group, then by D.I.Zaitzev theorem [8], D contains an < x >-invariant abelian subgroup Jo u rn al A lg eb ra D is cr et e M at h .38 On modules over group rings of locally soluble groups B = Drn∈N < bn > and we may assume that bn has prime order, for each n ∈ N. Let 1 6= c1 ∈ B and C1 =< c1 ><x>. Then C1 is finite and there is a subgroup E1 such that B = C1 ×E1. Let U1 = core<x>E1. Then U1 has finite index in B. If 1 6= c2 ∈ U1 and C2 =< c2 ><x>, then C2 is a finite < x >-invariant subgroup and < C1, C2 >= C1 × C2. Continuing in this manner, we can construct a family {Cn|n ∈ N} of finite < x >- invariant subgroups of B such that {Cn|n ∈ N}= Drn∈NCn. Lemma 2.1 implies that x ∈ L. The contradiction. The lemma is proved. From lemmas 2.5 and 3.1 validity of the theorem is followed. Theorem 3.2. Let A be Zp∞G-module, G is a locally soluble group which satisfies the condition min−nad, and the cocentralizer of G in module A is not Artinian Zp∞-module. Then either G satisfies the minimal condition or G = AD(G). Lemma 3.3. Let A be Zp∞G-module, G is a locally soluble group which satisfies the condition min − nad, and the cocentralizer of G in module A is not Artinian Zp∞-module. Then either G is soluble or G has an ascending series of normal subgroups 1 = W0 ≤ W1 ≤ ... ≤ Wn ≤ ... ≤ Wω = ∪n∈NWn ≤ G, such that the cocentralizer of each subgroup Wn in module A is Artinian Zp∞-module and Wn+1/Wn is abelian for n ≥ 0. Moreover, in this case G/Sω is soluble. Proof. At first we show that G is hyperabelian. It is sufficiently to show that every non-trivial image of G contains a non-trivial normal abelian subgroup. Let H be a proper normal subgroup of G. Suppose first that the cocentralizer of H in module A is not Artinian Zp∞-module. Then the quotient group G/H has the minimal condition on subgroups. Therefore it is Chernikov group and has a non-trivial abelian normal subgroup. Now we suppose that the cocentralizer of H in module A is Artinian Zp∞-module. Let S = {Mσ/H|σ ∈ Σ} be a family of all non-trivial normal subgroups of G/H. Suppose at first that for each σ ∈ Σ the cocentralizer of Mσ in module A is not Artinian Zp∞-module. We show that in this case G/H has the minimal condition on normal subgroups. Let {Mδ/H} be a non-empty subset of S. For any δ the cocentralizer of Mδ in module A is not Artinian Zp∞-module. By the condition min−nad the set {Mδ} has the minimal element M . Hence M/H is the minimal element of subset {Mδ/H}. Therefore G/H has the minimal condition on normal subgroups. Hence G/H is hyperabelian, and G/H has a non- trivial abelian normal subgroup. In the case where for some γ ∈ Σ the cocentralizer of Mγ is Artinian Zp∞-module, the subgroup Mγ is Jo u rn al A lg eb ra D is cr et e M at h .O. Yu. Dashkova 39 soluble. Hence Mγ/H is non-trivial normal soluble subgroup of G/H. Therefore G/H contains non-trivial normal abelian subgroup and so G is hyperabelian. Let 1 = H0 ≤ H1 ≤ ... ≤ Hα ≤ ... ≤ G be a normal ascending series with abelian factors and let α be the least ordinal such that cocentralizer of Hα in module A is not Artinian Zp∞-module. Then, as above, Hβ is soluble for all β < α. Moreover, the quotient group G/Hα has the minimal condition on subgroups and so it is Chernikov group. Suppose first that α is not a limit ordinal. Therefore Hα is soluble and it follows that G is soluble. Suppose now that α is a limit ordinal and that G is not soluble. For each positive integer d there exists an ordinal βd such that βd < α, Hβd has derived length at least d. Moreover, we may assume that βi < βi+1 for all positive integers i. For each positive integer i, let Ti = Hβi . So the group G has an ascending series of normal subgroups 1 = T0 ≤ T1 ≤ ... ≤. Now Tω = ∪n∈NTn is not soluble and so Tω = Hα. A series 1 = W0 ≤ W1 ≤ ... ≤ Wn ≤ ... ≤ Wω = ∪n∈NWn ≤ G with the properties referred to in the theorem can now be obtained by refining the series 1 = T0 ≤ T1 ≤ ... ≤ Tω ≤ G. The lemma is proved. Lemma 3.4. Let A be Zp∞G-module, G satisfies the condition min − nad, the cocentralizer of G in module A is not Artinian Zp∞-module, and G = AD(G). Then G/Gℑ is finite. Proof. Let us suppose for a contradiction that G/Gℑ is infinite. Then G has an infinite descending series of normal subgroups G ≥ N1 ≥ N2 ≥ ..., such that the quotient groups G/Ni are finite for each i. It follows that, for some k, G/Nk is finite and the cocentralizer of Nk in module A is Artinian Zp∞-module. Since G = AD(G), it can be choose the subgroup H such that H = AD(H), and G = HNk. Therefore the cocentralizer of G in module A is Artinian Zp∞-module. The contradiction. The lemma is proved. Lemma 3.5. Let A be Zp∞G-module, G satisfies the condition min − nad, and the cocentralizer of G in module A is not Artinian Zp∞-module. Suppose that G has an ascending series of normal subgroups 1 = W0 ≤ W1 ≤ ... ≤ Wn ≤ ... ≤ ∪n≥1Wn = G such that the cocentralizer of each subgroup Wn in module A is Artinian Zp∞-module and each Wn+1/Wn is abelian. Then G is soluble. Proof. Since A/CA(Sk) is Artinian Zp∞-module, as in the proof of lemma 2.5 we conclude that the additive group of the module A/CA(Sk) is Jo u rn al A lg eb ra D is cr et e M at h .40 On modules over group rings of locally soluble groups Chernikov group. Therefore there is a finite series of Zp∞G-submodules A = A0 ≥ A1 ≥ ... ≥ An(k) = CA(Sk), each factor of which is either finite Zp∞G-module or quasifinite Zp∞G-module. Since the cocentralizer of Sk+1 in module A is Artinian Zp∞-module the above series is extended to a series of Zp∞G-submodules A = A0 ≥ A1 ≥ ... ≥ An(k) ≥ ... ≥ An(k+1) = CA(Sk+1), each factor of which is either finite Zp∞G-module or quasifinite Zp∞G-module. In this way we obtain an infinite descending chain of Zp∞G-submodules A = A0 ≥ A1 ≥ A2 ≥ ... ≥ Aω = CA(G), each factor of which is either finite Zp∞G-module or quasifinite Zp∞G- module. Let H = ∩j≥0CG(Aj/Aj+1). By lemma 16.19 [1] for each j the quo- tient group G/CG(Aj/Aj+1) is abelian-by-finite. Since G/H embeds in the Cartesian product of the groups G/CG(Aj/Aj+1), it follows that G/H is abelian-by-(residually finite). Moreover, G is a union of subgroups such that their cocentralizers in module A are Artinian Zp∞-modules. There- fore G = AD(G). By lemma 3.4 the quotient group G/H is abelian-by- finite. Let K/H be a normal abelian subgroup of G/H such that G/K is finite. Since G = AD(G) the cocentralizer of K in module A is not Ar- tinian Zp∞-module. Let the cocentralizer of H in module A be Artinian Zp∞-module. Since the subgroup H is locally soluble, then the quotient group H/CH(A/CA(G)) is isomorphic to a locally soluble subgroup of GL(r,Zp∞). Since Zp∞ is an integral ring then it can be imbeded in the field F . Therefore the quotient group H/CH(A/CA(H)) is isomorphic to some locally soluble subgroup of linear group GL(r, F ). By corollary 3.8 [7] H/CH(A/CA(H)) is a soluble group. Since the quotient group CH(A/CA(H)) is a subgroup of stabilizer of the series of length 2, and therefore it is abelian. Hence H is a soluble group. Thus, we may suppose that the cocentralizer of H in module A is not Artinian Zp∞-module. We show that H is soluble. Let Lj = CH(A/Aj), j = 1, 2, ... . If H 6= Lj for some j then there exists the number t for which the quotient group H/Lt is infinite. Therefore there exists the number k ≥ j, k ≥ t, for which among the factors of the series A/Ak = A0/Ak ≥ A1/Ak ≥ A2/Ak ≥ ... ≥ Aj/Ak ≥ ... ≥ Ak/Ak there are infinite. By the results of chapter 8 [9] H has a nilpotent non-periodic image. Then there exists the normal subgroup H1 of H for which the quotient group H/H1 is a nilpotent non-periodic group. It follows that there is a normal subgroup H2, for which the quotient group H/H2 is an abelian torsion-free group, which contradicts lemma 2.4. Then H = Lj for each j, j = 1, 2, ... . Finally, suppose that for each j, j = 1, 2, ..., the quotient group H/Lj is finite. Suppose that there is a number j, for which the cocentralizer of Lj in module A is Artinian Zp∞-module. Let j be the minimal with this property. Therefore the cocentralizer of Lj−1 Jo u rn al A lg eb ra D is cr et e M at h .O. Yu. Dashkova 41 in module A is not Artinian Zp∞-module. Since Lj−1/Lj is finite and G = AD(G), then the cocentralizer of Lj−1 in module A is Artinian Zp∞- module. Contradiction. Therefore the cocentralizer of each subgroup Lj in module A is not Artinian Zp∞-module. Since H has the condition min − nad, there is the number m for which Lj = Lm for each j ≥ m. Therefore the subgroup Lm is soluble. Since H/Lm is finite then H is a soluble group also. The lemma is proved. From these results it follows the validity of theorem 1.1. Proof of theorem 1.2. By theorem 1.1 G is a soluble group. To prove the theorem it is sufficient to consider the case when G is not Chernikov group. Let G = D0 ≥ D1 ≥ D2 ≥ ... ≥ Dn = 1 be the derived series of G. By lemma there is the number m such that the cocentralizer of Dm in module A is not Artinian Zp∞-module while the cocentralizer of Dm+1 in module A is Artinian Zp∞-module. By lemma 2.4 the quotient groups Di/Dj+1, i = 0, 1, ..., m, are Chernikov groups. Let U = Dm+1. Then G/U is Chernikov group. Let C = CA(U). Then C is Zp∞G-submodule of module A. Since the cocentralizer of U in module A is Artinian Zp∞- module then A/C is Artinian Zp∞-module. Therefore there exists the series of submodules 0 = C0 ≤ C = C1 ≤ C2 ≤ ... ≤ Ct = A, such that each factor Ci+1/Ci, i = 1, ..., t − 1, is either finite Zp∞G- module or quasifinite Zp∞G-module. Then by lemma 16.19 [1] the quo- tient groups G/CG(Ci+1/Ci), i = 1, ..., t − 1, are abelian-by-finite. Since G/U is Chernikov group and U ≤ CG(C1), then G/CG(C1) is Chernikov group also. Then the quotient group G/CG(C1) is abelian-by-finite. Let H = CG(C1) ∩ CG(C2/C1) ∩ ... ∩ CG(Ct/Ct−1). It should be noted that G/H is an abelian-by-finite group. Let V/H be a normal abelian sub- group of G/H such that G/V is finite. By theorem 3.2 the cocentralizer of V in module A is not Artinian Zp∞-module. By lemma 2.4 V/H is Chernikov group. Therefore G/H is Chernikov group also. The sub- group H acts trivially on each factor of the series 0 = C0 ≤ C = C1 ≤ C2 ≤ ... ≤ Ct = A. Therefore H is a nilpotent group. The theorem is proved. Proof of theorem 1.3. By the conditions of the theorem G is not finitely generated group. We prove that G has not proper subgroups of finite index. Let N ≤ G and index |G : N | is finite. Then the finitely generated subgroup M can be chosen for which G = MN . M и N are Jo u rn al A lg eb ra D is cr et e M at h .42 On modules over group rings of locally soluble groups proper subgroups of G then their cocentralizers in module A are Artinian Zp∞-modules. Therefore the cocentralizer of G in module A is Artinian Zp∞-module. Contradiction. From theorem 1.2 the group G contains the normal nilpotent subgroup H such that the quotient group G/H is Chernikov group. If the quotient group G/H is finite then G is nilpotent- by-finite. Since G is not finitely generated then the subgroup H is not finitely generated also. Hence G/H ′ is infinitely generated abelian-by- finite group. Then G/H ′ = (G1/H ′)(G2/H ′), where G1 and G2 are proper subgroups of G. Contradiction. Therefore the quotient group G/H is a divisible periodic abelian group. If G/H is not isomorphic to Cq∞ for some prime q then G = G1G2, where G1 and G2 are proper subgroups of G. Contradiction. Therefore G/H is isomorphic to Cq∞ for some prime q. Hence there exists the ascending series of normal subgroups of G E ≤ H ≤ G1 ≤ G2 ≤ ... ≤ Gn ≤ ..., such that the quotient groups G1/H, Gi+1/Gi, i = 1, 2, ... are finite of prime order q and G/H = ⋃ n≥1(Gn/H). In view of the construction of additive group of Artinian Zp∞-module, for each n = 1, 2, ..., the module A has the series of G-invariant submodules 0 ≤ CA(Gn) ≤ A1 ≤ A2 ≤ ... ≤ Ak ≤ ..., such that each factor Ak/CA(Gn), k = 1, 2, ..., is finite. Hence the quo- tient groups G/CG(Ak/CA(Gn) are finite for each k = 1, 2, .... Previously it was proved that G has not proper subgroups of finite index. Hence, G = CG(Ak/CA(Gn) for each k = 1, 2, ..., and therefore [Ak, G] ≤ CA(Gn) for each k = 1, 2, .... Then [A, G] ≤ CA(Gn). Since this inclusion is fulfilled for each n = 1, 2, ..., then [A, G] ≤ ⋂ n≥1 CA(Gn) = CA(G). Hence G acts trivially on each factor of the series 0 ≤ CA(G) ≤ A. Therefore G is an abelian group. Let G be not isomorphic to Cq∞ for some prime q. The G is the product of two its proper subgroups. Therefore the cocentralizer of G in module A is Artinian Zp∞-module. Contradiction. Hence G ≃ Cq∞ for some prime q. The theorem is proved. References [1] Kurdachenko L.A., Otal J., Subbotin I.Ya. Artinian modules over group rings. Birg Häuser; Basel, Boston, Berlin.– 2007.– 248 p. [2] Dixon M.R., Evans M.J., Kurdachenko L.A. Linear groups with the minimal condi- tion on subgroups of infinite central dimension. Journal of Algebra. – 2004.– V.277, N1. – P.172-186. [3] Robinson D.J.R. Finiteness Conditions and Generalized Soluble Groups. Ergeb- nisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, Heidelberg, New York.– 1972.– V.1,2.– 464 p. Jo u rn al A lg eb ra D is cr et e M at h .O. Yu. Dashkova 43 [4] Kurdachenko L.A., Subbotin I.Ya., Semko N.N. Insight into Modules over Dedekind Domains. National Academy of Sciences of Ukraine, Institute of Math- ematics, Kiev.– 2008.– 117 p. [5] Kurosh A.G. Theory of groups. Moscow, Nauka. – 1967.– 648 p. [6] Merzliakov Yu.I. Rational groups. Moscow, Nauka. – 1980.– 464 p. [7] Wehrfritz B.A.F. Infinite Linear Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, New York, Heidelberg, Berlin.– 1973.– 229 p. [8] Zaicev D.I. On solvable subgroups of locally solvable groups. Dokl. Akad. Nauk SSSR. – 1974. – V.214, N6.– P.1250-1253. [9] Fuchs L. Infinite Abelian Groups. Moscow, Mir. – 1973. – V.1. – 229 p. Contact information O. Yu. Dashkova Department of Mathematics and Mechanics, Kyev National University, ul.Vladimirskaya, 60, Kyev, 01033, Ukraine E-Mail: odashkova@yandex.ru Received by the editors: 22.03.2009 and in final form 30.04.2009.

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