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The Extension of Results Due to Gorchakov and Tomkinson from FC-Groups to CC-Groups
JOURNAL OF ALGEBRA ARTICLE NO.
185, 314]328 Ž1996.
0327
The Extension of Results Due to Gorchakov and Tomkinson from FC-Groups to CC-Groups Miguel Gonzalez and Javier Otal Departamento de Matematicas, Facultad de Ciencias, Uni¨ ersidad de Zaragoza, ´ 50009 Zaragoza, Spain Communicated by Peter M. Neumann Received June 27, 1995
1. INTRODUCTION This paper is a contribution to the theory of embeddings of certain types of groups, continuing the work done in w4, 5x and announced in w6x Žone says that a group H embeds in the group K or that there is an embedding of H in K, if there exists an isomorphism between H and a subgroup of K .. We consider groups with finite conjugacy classes, or FC-groups. In the theory of FC-groups a classical problem introduced by P. Hall w9x was that of embedding residually finite periodic FC-groups as subgroups of direct products of finite groups. Since then, his work has been continued in a sequence of papers, such as those of Gorchakov and Tomkinson Žsee w17x for a complete account of this subject.. Gorchakov showed that if the periodic FC-group G is a subgroup of a cartesian product if isomorphic finite groups, then G can be embedded in a direct product of finite groups. Using this result, Tomkinson w16x characterized the residually finite periodic FC-groups as subgroups of centrally restricted products of finite groups. The groups we are interested in are groups with Cherniko¨ conjugacy classes, or CC-groups, which were introduced by Polovickiı˘ w13, 14x as an extension of the concept of FC-group, A group G is said to be a CC-group if GrCG Ž x G . is a Chernikov group, for each x g G. In w4x, we obtained some results on embeddings of residually Chernikov CC-groups, which generalise some of the known results for FC-groups. Some of these results will now become consequences of the results we obtain here, which are natural extensions of the results of Gorchakov and Tomkinson mentioned above. Although we are not able to obtain the full extension of these 314 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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results, our theorems apply in the most important situations and we are still able to improve our previous results. Our group-theoretic notation is standard and follows w2, 15, 17x. We shall use Polovickiı’s ˘ theorem characterizing CC-groups w15, Theorem 4.36x, which says that, if G is a CC-group, then the normal closure x G is Chernikov-by-cyclic and w G, x x is Chernikov, for every x g G. P. Hall’s notation of group-classes and closure operations Žsee w15x, for example. is useful. Thus C, F, and A 0 will denote respectively the classes of Cherniko¨ , finite, and torsion-free abelian groups, while S, Q, D, and R will be the usual closure operations. An X-residual system Ni : i g I 4 of a group of G is simply a set of normal subgroups Ni of G with trivial intersection such that GrNi g X, for all i g I. If G has an X-residual system then it is residually X or G g RX. If Gi : i g I 4 is a family of groups, G F Ł Gi : i g I 4 and J : I, we shall denote the projection of G under the J-fold coordinates, that is, its image under the natural mapping G ª Ł Gi : i g J 4 , by GŽ J .. If a s Ž a i . g ŁGi , then the support of a is the set supp a s i g I : a i / 14 . If S : ŁGi , the support of S is defined to be the union suppŽ S . of the supports of the elements a g S. If h is an infinite cardinal, we write Ł h Gi as the subgroup of the cartesian product ŁGi consisting of all elements whose support has cardinal strictly less than h ; the usual direct product DrGi of the family is the particular case h s / 0 . We denote by Ch the operation which associates to a class of groups X the class Ch X of groups of the form Ł h Gi , where the Gi are X-groups. Finally, we deal with the extension of Tomkinson’s definition of the centrally restricted product of finite groups w17, p. 29x to the centrally complete product of the Cherniko¨ groups Gi : i g I 4 , denoted by Zr*Gi , which is the subgroup of the cartesian product consisting of all elements with only finitely many non-central components. Its torsion subgroup is the centrally restricted product, denoted by ZrGi . Also, we put ZrhU Gi s ŽŁ Zi .ŽŁ h Gi . and Zrh Gi the torsion subgroup of ZrhU Gi . We shall be considering groups which are subgroups of Zr*Gi and ZrGi with Gi Chernikov and, as in w5x, we use the notation G g Zr*C and G g ZrC to indicate that G can be embedded in this way.
2. PRELIMINARY RESULTS Although we have not been able to extend Gorchakov’s theorem as completely as we wish, we have been able to establish a version of that and Tomkinson’s theorem for CC-groups having a Chernikov residual system which enjoys certain countability properties related to subgroups of the form X G , where X is a finite subset of G.
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We say that a group G is a c-residually Cherniko¨ group if it has a Chernikov residual system Ni : i g I 4 such that, for every finite subset X of G, X G l Ni : i g I 4 is a countable set. In this case Ni : i g I 4 is said to be a Ž Cherniko¨ . c-system of G. Clearly, a c-residually Chernikov group is residually Chernikov and we leave as an open question whether every residually Chernikov CC-group has a c-system. LEMMA 1. group G.
Let Ni : i g I 4 be a c-system of the residually Cherniko¨
Ž1. If H F G, then H l Ni : i g I 4 is a c-system of H. Ž2. If J : I and N s F Nj : j g J 4 , then NjrN : j g J 4 is a c-system of GrN. Proof. It is an immediate consequence of the definitions. We recall that, in general, a class of groups with the form RX is not Q-closed. This is the reason why we study properties such as Ž2. above, whose practical use is the following: Let Ni : i g I 4 be a c-system of the group G and Gk s GrNk , for every k g I, so that G F ŁGk . Then GŽ J . admits the family GŽ J . l ŽŁ Gk : k g I, k / j4 : j g J 4 as a c-system, where J : I. The next results are very important for our purposes of classification. They provide some theoretical examples of c-residually Chernikov groups. THEOREM 1. For an arbitrary CC-group G, the factor group GrZŽ G . is always a c-residually Cherniko¨ group. Proof. If Z s ZŽ G ., we shall show that CG Ž g G .rZ : g g G4 is a c-system of GrZ. We clearly may write the normal closure of any finite subset of GrZ in the form X G ZrZ, where X is a finite subset of G. Call K s X G , for a given X. If g g G, then it is clear that we have KZrZ l CG Ž g G .rZ s CK Ž g G . ZrZ, so it is enough to show that CK Ž g G . : g g G4 is a countable set for every K, as above. The factor group GrCG Ž K . is Chernikov and so it is countable. Let Y be a countable transversal of G modulo CG Ž K .. If g g G, we may write g s yz, where y g Y and z g CG Ž K .. Then we easily check that CK Ž g G . s CK Ž y G ., and thus the sets CK Ž g G . : g g G4 and CK Ž y G . : y g Y 4 have the same cardinality, so they are countable. A Prufer p-group is an example of a residually Chernikov FC-group ¨ which is not a residually finite group. Thus, the class of residually Chernikov FC-groups lies properly between those of residually finite FC-groups and of residually Chernikov CC-groups but groups of this type do have a c-system, as our next result shows.
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A residually Cherniko¨ FC-group is a c-residually Cherniko¨
Proof. This is clear since if X is a finite subset of the FC-group G, then X G is a finitely generated FC-group and so has only countably many subgroups Žsee w10x..
3. EXTENSION OF GORCHAKOV’S THEOREM TO CC-GROUPS Before dealing with the extension of Gorchakov’s theorem, we need some auxiliary results, which we shall show now. LEMMA 2. Let H be a subgroup of the Cherniko¨ group G. If R is the radicable part of G Ž see w11x, then R s w R, H xCR Ž H .. Proof. Since the subgroup H l R contains the radicable part of H, it has finite index in H. Then H s Ž H l R . F, where F is finite. By w15, Lemma 3.29.1x, R s w R, F xCR Ž F .. Note that R is abelian, so w R, F x s w R, H x, CR Ž F . s CR Ž H .. LEMMA 3. Let Fi : i g I 4 be a family of isomorphic Cherniko¨ groups and let h be an infinite cardinal such that h F < I <. If Z s Ł ZŽ Fi . and D s Ł h Fi , then D is a direct factor of ZD, that is, ZD s D = A, where A is an abelian group. Consequently Zr*Fi g SC / 1C, if h s / 0 , and ZrhU Fi g SCC, if h G / 1. Proof. For each i g I, we put ZŽ Fi . s A i = Bi , where A i is a finite group Bi is a radicable Chernikov group, that is, an abelian Chernikov group. Let A s Ł A i and B s Ł Bi . It is clear that Ž A l D . l A n s A n l D s Ž A l D . n for every n G 1, so A l D is a pure subgroup of A. Since A l D has a finite exponent, it is a direct factor of A, that is, A s Ž A l D . = E Žsee w2, Theorem 27.5x.. Suppose that b s ad, where a g A, b g B, and d g D. Thus
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The proof of Gorchakov’s theorem that we carry out below is based on the original as given in w17x. Despite this, our proof has required several changes; for example, the role played by the centre ZŽ G . in the FC-case is taken by a larger subgroup here. This further changes in the computations. THEOREM 3 ŽGorchakov’s theorem.. Let Ni : i g I 4 be a c-system of the CC-group G such that GrNi ( GrNj , for any pair of indices i, j g I. Then G g SC / I C. Proof. Put Fi s GrNi , for every i g I, so that we may assume that G F Ł Fi , where the Fi are isomorphic Chernikov groups. We shall show the result by induction as follows. Since each Fi g MIN, we may assume that whenever H F Ł Ej : j g J 4 is a CC-group having a c-system with the form H l Ker p j : j g J 4 , where J : I, the Ej are isomorphic groups, and each Ej has an isomorphic copy properly contained in Fj , then H g SC / 1C. To do that, we shall split the proof into several steps. For each i g I, let R i s RŽ Fi . be the radicable part of Fi . We consider R s Ž R i ZŽ Fi . : i g I 4 . R is abelian and GrR is an FC-group, because Fpy1 i RŽ G . F R w11, Lemma 2.1x. Put G s GrR and Fis FirR i ZŽ Fi ., i g I. Then GŽ i . s Fi, i g I, G F ŁFi, and, if J : I, GŽ J . ( GŽ J .rRŽ J .. J is called homogeneous if, for every j g J, the map p j: GŽ J . ª Fj is an isomorphism. Clearly, any subset of homogeneous set is also homogeneous. Concerning homogeneous sets, we quote the following result Žsee w17x.. Ža. E¨ ery homogeneous subset of I is contained in a maximal one and two different maximal homogeneous subsets of I are disjoint. By Ža. above and the fact that any atomic subset of I is homogenous, one has that the different maximal homogeneous subsets of I form a partition of I. They are called the homogeneous components for G in I. The next fact actually shows that these components are well behaved under projections w17, Lemma 2.16x. Žb. If Jl : l g L4 are the homogeneous components for G in I and J : I, then the non-empty members of the set Jl l J : l g L4 are the homogeneous components for GŽ J . in J. The next result describes the structure that a homogeneous component determines on its corresponding projection. Žc. Let J be a homogeneous component for G in I. Then we ha¨ e the normal decomposition GŽ J . s HL, that is, H and L are normal subgroups of GŽ J ., where L is an abelian group and H is the normal closure in GŽ J . of a finite set. In particular H is Cherniko¨ -by-Ž finitely generated abelian. and countable. Moreo¨ er there exists a countable subset K of J such that H ( H Ž K ..
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Since J is homogeneous, we have that GŽ J . ( Fj, for every j g J. Each factor FjrR j is finite, and so is GŽ J .. But GŽ J . ( GŽ J .rRŽ J ., so GŽ J . s RŽ J . H, where H is the normal closure in GŽ J . of a finite set and we note that L s RŽ J . is abelian. Therefore the first assertion of Žc. is proved. Moreover, GŽ J . s GrŽF Ker a j : j g J 4., where a j : G ª Fj is the canonical map, j g J. Also, if j g J and a j: GŽ J . ª Fj denotes the natural projection, since H l ŽF Ker a j : j g J 4. s 1, for every 1 / h g H, there is j s jŽ h. g J such that h f Ker a j. If K s jŽ h. : 1 / h g H 4 : J, K is countable and H l ŽF Ker a j : j g K 4. s 1. Thus H ( H Ž K ., which shows Žc.. To apply the induction, we need the following concept. A subset J of I is said to be a reduced subset of I if one can embed the group GŽ J . in a cartesian product of isomorphic Chernikov groups and these groups are isomorphic to proper subgroups of the Fi . If J is the union of the family Jv : v g V 4 of reduced sets, we may write J as a disjoint union of reduced sets and, bearing in mind that GŽ J . is isomorphic to a subgroup of ŁGŽ Jv ., it follows that J is also a reduced set. Therefore I contains a unique maximal reduced set, J say, which is called the reduced component for G in I. In this case, I y J contains no reduced sets for G. If J is the reduced component for G in I, we assume that GŽ J . F Ł Ej : j g J 4 , and Ej - Fj for all j g J. It is known that there exists only a countable number of isomorphism types of Chernikov groups, Cn : n g N4 say ŽWehrfritz w18x.. If n g N, let Jn s j g J : Ej ( Cn4 ; then J s D Jn , GŽ J . F ŁGŽ Jn ., and GŽ Jn . F Ł Ej, n , where Ei, n ( Cn for all j g Jn . By our induction, we may assume that GŽ Jn . g SC / 1C. Then GŽ J . g SC / 1C and it is enough to show that GŽ I y J . g SC / 1C. Therefore, we suppose that I contains no non-empty reduced subsets and we must show that GŽ I . g SC / 1C. By Ža., we write I as a disjoint union of its homogeneous components, I s D Jl : l g L4 say. For each l g L, we consider the normal decomposition GŽ Jl . s Hl Ll , where Hl is a Chernikov-by-Žfinitely generated abelian . normal subgroup, Ll is an abelian normal subgroup, and Hl ( HlŽ Kl . with Kl countable. If l g L, let Ml s F Ker p j : j g Kl4 so that GrMl ( GŽ Kl .. Let Nl be the inverse image in G of Hl under p Jl. Then Ml l Nl s F Ker p j : j g Jl4 and GrŽ Ml l Nl . ( GŽ Jl .. Observe that G can be embedded in Ž GrŽF Ml .. = Ž GrF Nl .., because ŽF Ml . l ŽF Nl . s 1. But GrNl ( GŽ Jl .rHl s Ll HlrHl is abelian, and so is GrŽF Nl .. By w5, Corollary 1x, GrŽF Nl . g SC / 1C. We now consider the group GrŽF Ml . ( GŽ K ., where K s D Kl . It is clearly sufficient to show that GŽ K . g SC / 1C. By Žb., the countable sets Kl are the homogeneous components for GŽ K . in K. Also, a non-empty reduced set for GŽ K . in K is reduced for G in I; so K has no non-empty reduced subsets. If GŽ K . is replaced by G and K by I the proof of the theorem will be completed
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when we establish Žd. If G F Ł Fi : i g I 4 , the homogeneous components of I are countable sets and G contains some element which has an uncountable number of non-central coordinates, then I contains an uncountable reduced subset. Throughout, we keep the above conditions together with the established notation. We split the proof of Žd. into several steps. Step 1. If H 1 G and GrCG Ž H . is a Cherniko¨ group, then H Ž i . ZŽ Fi . is a proper subgroup of Fi for all but countably many indices i g I. Put J s i g I : H Ž i . ZŽ Fi . s Fi 4 , H1 s H Ž J ., G1 s GŽ J ., and Z j s ZŽ Fj ., j g J. Thus H 1 1 G 1 and Fj s H 1Ž j . Z j , j g J. C G 1Ž H 1 . s G 1 l ŽŁCF ŽŽ H1ŽŁ Z j ..Ž j ... s G1 l ŽŁ Z j . and G1rCG Ž H1 . is Chernikov. Put j 1 R1 s G1 l ŽŁ Z j R j ., so that G1rR1 F ŁŽ FjrZ j R j . and then G1rR1 is residually finite. But G1rR1 is also Chernikov, so it is finite and has finitely many subgroups, L1 , . . . , L s say. If u j : G1rR1 ª FjrZ j R j , j g J, is the projection and Jr s j g J: Ker u j s L r 4 , then there are isomorphism maps Ž G1rR1 .Ž Jr . ª FjrZ j R j , j g Jr , and therefore each Jr is a homogeneous set for G1 in J and for G in I. Hence, by hypothesis, Jr is countable. Therefore J is countable. Given J : I, we add to the above the following notation Ždepending on J .: If A F Ł Fj : j g J 4 , we shall write A s AZrZ, where Z s Ł Z j : j g J 4. Step 2. We now suppose that I contains an uncountable subset, J say, such that G Ž J . F ŽŁ / 1 Fj : j g J 4. H, where H is the normal closure in GŽ J . of a finite set and H Ž j . / 1 for e¨ ery j g J. Then J contains an uncountable reduced set for G. Clearly we may assume that J s I. Put L s GŽŁ Zi ., so that Z s Ł Zi F ZŽ L.. In fact, we have Z s ZŽ L., because the corresponding projections are onto maps. Since L s GZ, we have that L s G F PH, where P s Ł / 1 Fi. Thus L s Ž L l P . H. Let E be the inverse image of L l P under the projection of L in L. If g g L, there are x g E and y g H such that the image under the above projection of g Ž xy .y1 is trivial, so g Ž xy .y1 g Z. Thus g s xyz s Ž xz . y, where z g Z, and we note that xz g E. Then L F EH, and so L s EH. Similarly, one can show that E s DZ, where D s L l ŽŁ / 1 Fi . s E l ŽŁ / 1 Fi .. Now GrCG Ž H . is a Chernikov group and, by Step 1, K 1 s i g I: H Ž i . Zi s Fi 4 is countable. Moreover L s EH s DZH and we find that D l H F Ł / 1 Fi is countable. Therefore, it follows that K 2 s suppŽ D l H . is countable. Put K s I y Ž K 1 j K 2 .. Then LŽ K . s D Ž K . H Ž K . ZŽ K . and D Ž K . s LŽ K . l ŽŁ / 1 Fk : k g K 4.. DŽ K . l H Ž K . s 1 and, since K is uncountable, we may again assume that I s K. In this case, we may write L s Ž D = H . Z and, if i g I, Zi - H Ž i . Zi and H Ž i . Zi - Fi .
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Put X s ZŽ H . and Y s ZŽ D . so that Ž D = H . l Z s Y = X. Write Z s ŁŽ A i = Bi ., where the A i are isomorphic finite groups and the Bi are isomorphic radicable Chernikov groups. Y s Ł / 1 Zi s Ł / 1 Ž A i = Bi . and, if A s Ł A i , B s Ł Bi , A1 s Ł / 1 A i and B1 s Ł / 1 Bi , Z s A = B, and Y s A1 = B1. We claim that there is a direct decomposition Z s Y = V = U, such that V is countable and X F Y = V s T. For, as in Lemma 3, A s A1 = A 2 and B s B1 = B2 . A 2 has a finite exponent and so it is a direct product of finite cyclic groups, A 2 s Dr C j : j g L4 say. As B2 is divisible, it is a direct product of some Prufer ¨ groups and some copies of the full rational group, B2 s Dr Di : k g K 4 say. Note that the groups C j and D k are countable. Since X is a countable subgroup of Z, there are countable subsets L9 of L and K 9 of K such that X F A1 = Ž Dr C j : j g L9.4 = B2 = Ž Dr Di : k g K 94.. If we put AX2 s Dr C j : j g L94 and B2X s Dr Di : k g K 94 such that A 2 s AX2 = AY2 and B2 s B2X = B2Y , then if V s AX2 = B2X and U s AY2 = B2Y , our claim has been carried out. Put W s XV, T s YW, and Z s YW = U. Thus L s DHZ s DSU, where S s HW, which is countable. Since w D, S x F w D, HZ x s w D, H x s 1, it follows that ZŽ DS . s ZŽ D . ZŽ S . s YXW, and so DS l U s YXV l U F T l U s 1 and L s DS = U. Put I1 s I y suppŽ D l S ., and let d g D, s g S, and u g U be elements such that dsuŽ I1 . s 1. Since suppŽ D l S . is countable, it follows that dsu g L l Ł / 1 Fi s D. Therefore, u g DS l U s 1, ds g D, s g D l S and it readily follows that sŽ I1 . s 1 and dŽ I1 . s 1. This shows that LŽ I1 . s DŽ I1 . = SŽ I1 . = UŽ I1 .. For every i g I1 , we put SŽ i . s H Ž i .W Ž i . F H Ž i . Zi . By hypothesis, H Ž i . g Zi . Since w DŽ i ., SŽ i .x s 1, DŽ i . - Fi . Since U is a subgroup of Z, we similarly have that UŽ i . - Fi . Thus I1 is the required reduced set, and Step 2 is completed. Step 3. The end of the proof of Žd. and of Theorem 3. It suffices to construct a J : I satisfying the conditions of Step 2. Suppose we may pick an x g G such that x Ž i . f Zi for an uncountable number of indices. Replacing this set by I, we may assume that x Ž i . f Zi for all i g I. If we Ž R i . : i g I 4 , then GrS F ŁŽ FirR i . where each factor define S s lpy1 i FirR i is finite with given order, n say. A subset T : G is said to be a parallel set for a non-empty subset J of I if, whenever g / h g T, in the above embedding, one has that Ž gS .Ž j . / Ž hS .Ž j . for every j g J. We clearly have that, if T is a parallel set of some J, then < T < F < FirR i < s n. Thus, by the choice of x, the set x, 14 is parallel for I, so, by the finiteness of n, we may choose a maximal parallel set T0 for I. If K is an uncountable subset of I, then T0 is a parallel set for K, and so is contained in a maximal parallel set for K. Choose an uncountable subset I0 of I and a maximal parallel set T for I0 such that T has largest possible order. Then T is a maximal parallel set for all uncountable subsets of I0 .
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Let T s x 1 , . . . , x t 4 . Since we can multiply the elements of T on the left by any element of G, we may assume that x f T. Now let H s ² x, x 1 , . . . , x t :T . Since G is a c-residually Chernikov group, H l Ker p i : i g I 4 is countable, and then there is J : I uncountable such that H l Ker p i s H l Ker p j for every pair i, j g J. Replacing J by I, we may assume that whenever h g H and hŽ i . s 1 for some i g I, then hŽ i . s 1 for every i g I, that is, h s 1. If g g G, then, for some 1 F j F t and .Ž . i g I, Ž gS .Ž i . s Ž x j S .Ž i ., or Ž xy1 j g i s r i g R i . By Lemma 2, R i s CR iŽ H Ž i ..w R i , H Ž i .x, so ri s c i h i , where c i g CR iŽ H Ž i .. and h i g w R i , H Ž i .x. Since w R i , H Ž i .x F H Ž i ., it follows that there is some h g H such that y1 .Ž . y1 xŽ . hŽ i . s h i , and then Ž xy1 i s c i . Therefore, w h, xy1 i s 1 and j gh j gh y1 y1 x w Ž . so H, x j gh s 1. That is, g g HCG H and G s HCG Ž H .. Thus CG Ž H .Ž i . - Fi , since x g H and x Ž i . f Zi , i g I. If CG Ž H . F Zr/ 1 Fi s ŽŁ Zi .ŽŁ / 1 Fi ., proceeding as in Step 2 and taking J s I, we show that G F ŽŁ / 1 Fi : i g I 4. H and we see that all the conditions are satisfied. Then the proof of Žd. is finished. We may now deduce from these facts the proof of Gorchakov’s theorem. If we assume that C s CG Ž H . g Zr/ 1 Fi , there is some c g C with an uncountable number of non-central coordinates. Put Ci s C Ž i . - Fi , i g I. Since the number of isomorphism types of Chernikov groups is countable, there is an uncountable subset I1 of I such that cŽ i . is not central and the Ci are isomorphic, i g I1. By Lemma 1 and by induction, there is an uncountable J : I such that C Ž J . F ŽŁ / 1 C j : j g J 4.H1 and the conditions of Step 2 are satisfied. But ZŽ C . s C l ZŽ G ., and, in particular C js C jrZŽ C j . s Ž CrZŽ C ..Ž j . F Ž GrZŽ G ..Ž j ., for every j g J. Thus G Ž J . F ŽŽŁ / 1 C j : j g J 4. H 1 . H Ž J . F ŽŁ / 1 Fj : j g J 4. ŽH1H .Ž J .. Since ŽH1H .Ž J . is the normal closure of some finite set and ŽH1H .Ž j . / 1, j g J, by Step 2, the proof is completed.
4. EXTENSION OF TOMKINSON’S THEOREM TO CC-GROUPS AND SOME OF ITS APPLICATIONS As an important consequence of Theorem 3, we may classify the c-residually Chernikov CC-groups as subgroups of centrally complete products of Chernikov groups, which is an extension of the CC-case of Tomkinson’s theorem w16x. THEOREM 4. For a group G, the following conditions are equi¨ alent. Ž1. G is a c-residually Cherniko¨ CC-group. Ž2. G g Zr*C. Further, if G is periodic, Ž2. can be replaced by Ž29. G g ZrC.
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Proof. Clearly, the second assertion is a consequence of the first one together with the obvious fact that a periodic Zr*C-group is a ZrC-group. Ž2. « Ž1. Let Gi : i g I 4 be a family of Chernikov groups. The classes of CC-groups and c-residually Chernikov groups are S-closed Žsee Lemma 1., so it suffices to show that the whole group G s Zr*Gi satisfies Ž1.. Therefore it is enough to prove that G in fact has a c-system. Let Cn : n g N4 be a representative set of isomorphism types of Chernikov groups. For every n G 1, put In s i g I : Gi ( Cn4 . By Lemma 3, we have that Zr* Gi : i g In4 g SC / 1C and, since I is the union of the In , G g SC / 1C also. Suppose that G F Ł / 1 Hj : j g J 4 , where the Hj are Chernikov. If p j : G ª Hj is the canonical projection and X : G is a finite set, suppŽ X G . is countable, while if j / suppŽ X G ., then X G F G l Ker p j . Therefore X G l Ker p j : j g J 4 is countable, and so it is in fact a c-system as desired. Ž1. « Ž2. Suppose first that G F Ł Gi : i g I 4 is a CC-group, where the Gi are Chernikov groups, each map p i : G ª Gi is onto, i g I, and G l Ker p i : i g I 4 is a c-system of G. As above, let Cn : n g N4 be a representative set of isomorphism types of Chernikov groups. For each n G 1, let In s i g I: Gi ( Cn4 . Then I is the union of the In , and G F GŽ I1 . = GŽ I2 . = . . . ,GŽ In . F Ł Gi : i g In4 , n G 1. By Lemma 1, GŽ In . is a CC-group admitting GŽ In . l Ker p i : i g In4 as a c-system. Then, by Theorem 3, GŽ In . g SC / 1C, and so G g SC / 1C. Thus, changing the notation if necessary, we may assume that G F Ł / 1 Gi : i g I 4 and show our result by induction on < I <. If I is countable, it follows from w4, Theorem 3.1x, so we may assume that I s a : a - r 4 is uncountable, where r is the least ordinal such that < r < s < I <; clearly, r has to be a limit ordinal. We shall show that I can be expressed as a disjoint union of certain subsets Ja , a - r , such that, if Ia s D Jb : b - a 4 , then the following conditions are satisfied. Ža. a g Ia j Ja . Žb. < Ia j Ja < F max / 0 , < a <4 . Žc. If we define Ha s G l ŽŁ Gg : g g Ia 4 , Ma s G l ŽŁ Gg : g f Ia 4 and Ca s CG Ž CrMa ., then one has that G s Haq1Ca , for every a - r . Given a - r , we assume defined every Jb , b - a , satisfying the conditions: Ža9. d g Id j Jd , for every d - a . Žb9. < Id j Jd < F max / 0 , < d <4 , for every d - a . Žc9. G s Hdq1Cd , for every d such that d q 1 - a . If a is a limit, we define Ja s a 4 , if a f Ia , and Ja s B, otherwise. Clearly each Ja is disjoint with the previous Jb , in such a way that the
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conditions Ža9. and Žb9. are satisfied for d s a , and the condition Žc9. is satisfied, for every d - a . If a s « q 1, then GrM« embeds in P Gd : d g I« 4 and, by w12, Lemma 6.3x, < GrC« < F max a 1 , < « <4 . Therefore, there is an S« F G such that G s S« C« and < S« < F max a 0 , < « <4 . Let K« be the set of all indices g such that the group S« admits a non-trivial projection on the component Gg . But, each element has countable support, so that < K« < F max a 0 , < « <4 . If Ja s Ž a 4 j K« . y Ia , then Ja and the Jb are disjoint and the conditions Ža9. and Žb9. hold for d s a . Further, G s Ha C« , so Žc9. holds for all d - a , and we may define Ja satisfying Ža9., Žb9., and Žc9. and hence Ža., Žb., and Žc.. If Na s G l ŽŁ Gb : b f Jaq1 4. and Da s CG Ž GrNa ., Maq1 F Na , G s Haq2 Caq1 F Haq2 Da , for all a . By w17, Lemma 2.20x, G F Zr*Ž GrNa .. Each GrNa embeds in Ł / 1 Gb : b g Jaq1 4 , so GrNa g Zr*C. Then, by w17, Lemma 2.23x, G g Zr*C. We deduce some consequences of Theorem 4 in order to classify some classes of groups. The next result improves w4, Corollary 5x Žsee also w5, Corollary 2x.. COROLLARY 1. For a CC-group G, we ha¨ e that GrZŽ G . g Zr*C. Furthermore, GrZŽ G . g ZrC pro¨ ided GrZŽ G . is periodic. Proof. It suffices to apply Theorems 1 and 4. We can now extend results first shown by Tomkinson w16x and Gorchakov w7x, in classifying residually Chernikov FC-groups. COROLLARY 2. For a group G, the following conditions are equi¨ alent. Ž1. G is a residually Cherniko¨ FC-group. Ž2. G is isomorphic to a subgroup of a centrally complete product of Cherniko¨ groups, which are central-by-finite groups. Further, if G is periodic, the embedding in Ž2. can in fact be into a centrally restricted product. Proof. As usual, the second assertion is an obvious consequence of the first one. Ž1. « Ž2. By Theorems 2 and 4, G g Zr*C. Also, we may assume that the corresponding components of G are Chernikov FC-groups. Therefore, they are also central-by-finite groups, claimed. Ž2. « Ž1. This is clear. COROLLARY 3. A residually Cherniko¨ FC-group is a QSDŽF j A 0 .group. It is a QSDF-group in the periodic case. Proof. Applying Corollary 2, we see that if G F Zr* Gi : i g I 4 , where the Gi are central-by-finite Chernikov groups, then g g QSDŽF j A 0 .. In fact, since the class QSDŽF j A 0 . is S-closed, we may assume that G s Zr*Gi s ZD, where Z s Ł ZŽ Gi . and D s DrGi , and show that G g
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QSDŽF j A 0 .. But an abelian group can be embedded in the direct product of a periodic abelian group and a torsion-free abelian group and, by w17, Corollary 3.4x, a periodic abelian group is a QSDF-group, so that Z g QSDŽF j A 0 .. If H is a central-by-finite Chernikov group, H s ZŽ H . F, with F finite, and, by w5, Lemma 1x, H g QSDŽF.. Then, since QSDŽF. is D-closed, D g QSDŽF.. Then, by w5, Lemma 1x, G g QSDŽF l A 0 .. The second assertion can be obtained by reasoning in a similar way since in this case Z is a periodic abelian group and so is a QSDF-group. Other special cases arise as nice particular consequences and as an example we develop some of them. The first needs an auxiliary result, which shows a property of countability that does work not in general. Although a Chernikov group is countable, it can contain an uncountable number of subgroups. Let A s ² a i : i G 1: and B s ² bi : i G 1: be two copies of a Prufer ¨ p-group and let G s A = B be their direct product. For each sequence S s Ž « 1 , « 2 , . . . . consisting of ‘‘zeros’’ and ‘‘ones’’, the subgroup of G given as G S s ² a1Ž b 1 . « 1 , a 2 Ž b 2 . « 1 Ž b 1 . « 2 , a3 Ž b 3 . « 1 Ž b 2 . « 2 Ž b1 . « 3 , . . . : is also a Prufer ¨ group and different sequences generate different subgroups of G, so that G has an uncountable number of subgroups. The above G is a radicable group and it is the direct product of two copies of the same Prufer ¨ group. However, if the radicable part of the Chernikov group G is the direct product of Žfinitely many. different Prufer ¨ groups, the situation is very different, as the next result shows. LEMMA 4. We suppose that the radicable part A of the Cherniko¨ group G is a locally cyclic group. Then G only has countably many subgroups. Proof. We may certainly assume that G is an infinite group. Since A is the unique maximal radicable subgroup of G, a subgroup H of G has the form H s BF, where B is a subgroup of A and F is a finite subgroup of G. Since G is countable, it has countably many finite subgroups F only because these are the finite parts of a countably infinite set. Therefore, it will be enough to see that A has only countable many subgroups. But A has the form A s A1 = ??? = A r , where A i is a group of type p`i and the pi are different primes. Thus, if B F A, since the A i have their elements with coprime order, we have that B s Ž B l A1 . = ??? = Ž B l A r .. As each A i has countably many subgroups, the number of subgroups of A is countable, as claimed. COROLLARY 4. Let G be a residually Cherniko¨ CC-group whose radicable part R has rank 1. Then G g Zr*C. Further G & ZrC, pro¨ ided G is periodic.
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Proof. If X : G is finite and T s T Ž X G ., then, by w1, Lemma 1x, T is Chernikov and X GrT is free abelian of finite rank. Thus, if H F X G , HrŽ H l T . is finitely generated, and hence H s Ž H l T . F, where F is finitely generated. X G is countable, and then it contains only a countable number of these F. By Lemma 4, T has a countable number of subgroups, and so has H. It follows immediately that G is a c-residually Chernikov CC-group, and so Theorem 4 applies. COROLLARY 5. Let G be a CC-group such that GrG9ZŽ G . has a finite exponent. Then GrZŽ G . g SDC. Proof. Since G9ZŽ G .rZŽ G . s Ž GrZŽ G ..9, it must be periodic. Also, GrG9ZŽ G . is periodic, and so is GrZŽ G .. By Theorems 1 and 4, GrZŽ G . g ZrC. Put L s GrZŽ G . and assume that L F ZrGi s ZD, where the Gi are Chernikov, Z s T ŽŁ ZŽ Gi .., and D s DrGi . If n G 1, let z s Ž z i . g Z l LD be such that z n g Z l D. Since suppŽ z n . s I0 is finite, if g s Ž g i ., where g i s z i , if i g I0 and g i s 1, otherwise, g g Z l D and z n s g n. Then Z l D is a pure subgroup of Z l LD. But Ž Z l LD .rŽ Z l D . ( Ž Z l LD . DrD s LDrD ( LrŽ L l D . is abelian and isomorphic to a quotient of LrL9 ( GrG9Z. Therefore Ž Z l LD .rŽ Z l D . has finite exponent and it follows that Z l D is a direct summand of Z l LD w2, Theorems 17.2, 28.2x, that is, Z l LD s Ž Z l D . = E, for a certain subgroup E. Therefore LD s Ž Z l LD . D s DE. But D l E s D l E l Z s 1, so we actually have that LD s D = E. By w5, Theorem 2x, E g SDC. Since D g SDC, we have that L g SDC. It is worth mentioning that there is an infinite extraspecial group G which is not a QSDC-group Žsee w17, Example 3.8x.. This group satisfies the hypothesis of Corollary 5 but not of w4, Theorem 5.3x. Therefore this corollary is not a consequence of the result given there.
APPENDIX It is an open question whether the analogue of Tomkinson’s theorem holds in the CC case, that is, whether or not a residually Chernikov CC-group is always a Zr*C-group or, equivalently, if those groups have a c-system. The aim of this appendix is to clarify the status of this question. In generalizing the result due to Gorchakov w7x, we showed in w4, Theorem 4.2x that a residually Chernikov CC-group is a QSD ŽC j A 0 .-group. In fact, Zr*C : QSDŽC j A 0 . w5, Corollary 2x, but this inclusion must be strict, since there are quotients of residually Chernikov groups that have no Chernikov residual systems. This example tells us that we cannot expect a full characterization of the latter groups to be given by this result.
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We may ask if there exists an embedding of any residually Chernikov CC-group in a centrally complete product of simpler groups, although they are not necessarily Chernikov groups. The answer to this question is affirmative, when we consider, for example, the countable CC-groups, which were classified in full in w3x. In fact if we make use of the continuum hypothesis, we may prove results such as the following. THEOREM 5. E¨ ery residually Cherniko¨ CC-group G embeds in a centrally complete product of countable CC-groups. Actually, this product can be considered centrally restricted, if the gi¨ en group G is periodic. The proof of this result can be carried out proceeding as in other parts of this paper and using other, more specific techniques, such as the continuum hypothesis. For these reasons, we shall give only a brief sketch of it. First, we shall need an auxiliary result. LEMMA 5. If G is a residually Cherniko¨ CC-group, then G embeds in a cartesian product of Cherniko¨ groups with
ACKNOWLEDGMENT Both authors are very grateful to Dr. Mike J. Tomkinson for his valuable suggestions and his hospitality at the Glasgow University, where many ideas of this work were carried out.
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REFERENCES 1. J. Alcazar and J. Otal, Slylow subgroups of groups with Chernikov conjugacy classes, J. Algebra 110 Ž1987., 507]513. 2. L. Fuchs, ‘‘Infinite Abelian Groups,’’ Academic Press, New York, 1970]1973. 3. M. Gonzalez and J. Otal, P. Hall’s covering group and embedding of countable CC-groups, Comm. Algebra 18 Ž1990., 3405]3412. 4. M. Gonzalez and J. Otal, Embedding theorems for residually Chernikov CC-groups, Proc. Amer. Math. Soc. 123 Ž1995., 2323]2332. 5. M. Gonzalez and J. Otal, Embeddings of central-by-Chernikov groups and central products of Chernikov groups, preprint. 6. M. Gonzalez and J. Otal, Embedding of CC-groups: A survey, preprint. 7. Yu. M. Gorchakov, Locally normal groups, Siberian Math. J. 12 Ž1971., 907]916. 8. Yu. M. Gorchakov, Theorems of Prufer]Kulikov type, Algebra and Logic 13 Ž1974., ¨ 374]378. 9. P. Hall, Periodic FC-groups, J. London Math. Soc. 34 Ž1959., 289]304. 10. B. H. Neumann, Groups with finite classes of conjugate elements, Proc. London Math. Soc. Ž 3 . 1 Ž1951., 178]187. 11. J. Otal and J. M. Pena, ˜ Characterizations of the conjugacy of Sylov p-subgroups of CC-groups. Proc. Amer. Math. Soc. 106 Ž1989., 605]610. 12. J. Otal, J. M. Pena, ˜ and M. J. Tomkinson, Locally inner automorphisms of CC-groups, J. Algebra 141 Ž1991., 382]398. 13. Ya. D. Polovickiı, ˘ On locally extremal groups and groups with the condition of pminimality, So¨ iet Math. Dokl. 2 Ž1961., 780]782. 14. Ya. D. Polovickiı, ˘ Groups with extremal classes of conjugate elements, Sibirsk. Mat. Zh. 5 Ž1974., 891]895. 15. D. J. S. Robinson, ‘‘Finiteness Conditions and Generalized Soluble Groups,’’ SpringerVerlag, Berlin 1972. 16. M. J. Tomkinson, A characterization of residually finite periodic FC-groups, Bull. London Math. Soc. 13 Ž1981., 133]137. 17. M. J. Tomkinson, ‘‘FC-Groups,’’ Pitman, London, 1984. 18. B. A. F. Wehrfritz, Sylov subgroups of locally finite groups with Min-p, J. London Math. Soc. (2) 1 Ž1969., 421]427.
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The Extension of Results Due to Gorchakov and Tomkinson from FC-Groups to CC-Groups Miguel Gonzalez and Javier Otal Departamento de Matematicas, Facultad de Ciencias, Uni¨ ersidad de Zaragoza, ´ 50009 Zaragoza, Spain Communicated by Peter M. Neumann Received June 27, 1995
1. INTRODUCTION This paper is a contribution to the theory of embeddings of certain types of groups, continuing the work done in w4, 5x and announced in w6x Žone says that a group H embeds in the group K or that there is an embedding of H in K, if there exists an isomorphism between H and a subgroup of K .. We consider groups with finite conjugacy classes, or FC-groups. In the theory of FC-groups a classical problem introduced by P. Hall w9x was that of embedding residually finite periodic FC-groups as subgroups of direct products of finite groups. Since then, his work has been continued in a sequence of papers, such as those of Gorchakov and Tomkinson Žsee w17x for a complete account of this subject.. Gorchakov showed that if the periodic FC-group G is a subgroup of a cartesian product if isomorphic finite groups, then G can be embedded in a direct product of finite groups. Using this result, Tomkinson w16x characterized the residually finite periodic FC-groups as subgroups of centrally restricted products of finite groups. The groups we are interested in are groups with Cherniko¨ conjugacy classes, or CC-groups, which were introduced by Polovickiı˘ w13, 14x as an extension of the concept of FC-group, A group G is said to be a CC-group if GrCG Ž x G . is a Chernikov group, for each x g G. In w4x, we obtained some results on embeddings of residually Chernikov CC-groups, which generalise some of the known results for FC-groups. Some of these results will now become consequences of the results we obtain here, which are natural extensions of the results of Gorchakov and Tomkinson mentioned above. Although we are not able to obtain the full extension of these 314 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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results, our theorems apply in the most important situations and we are still able to improve our previous results. Our group-theoretic notation is standard and follows w2, 15, 17x. We shall use Polovickiı’s ˘ theorem characterizing CC-groups w15, Theorem 4.36x, which says that, if G is a CC-group, then the normal closure x G is Chernikov-by-cyclic and w G, x x is Chernikov, for every x g G. P. Hall’s notation of group-classes and closure operations Žsee w15x, for example. is useful. Thus C, F, and A 0 will denote respectively the classes of Cherniko¨ , finite, and torsion-free abelian groups, while S, Q, D, and R will be the usual closure operations. An X-residual system Ni : i g I 4 of a group of G is simply a set of normal subgroups Ni of G with trivial intersection such that GrNi g X, for all i g I. If G has an X-residual system then it is residually X or G g RX. If Gi : i g I 4 is a family of groups, G F Ł Gi : i g I 4 and J : I, we shall denote the projection of G under the J-fold coordinates, that is, its image under the natural mapping G ª Ł Gi : i g J 4 , by GŽ J .. If a s Ž a i . g ŁGi , then the support of a is the set supp a s i g I : a i / 14 . If S : ŁGi , the support of S is defined to be the union suppŽ S . of the supports of the elements a g S. If h is an infinite cardinal, we write Ł h Gi as the subgroup of the cartesian product ŁGi consisting of all elements whose support has cardinal strictly less than h ; the usual direct product DrGi of the family is the particular case h s / 0 . We denote by Ch the operation which associates to a class of groups X the class Ch X of groups of the form Ł h Gi , where the Gi are X-groups. Finally, we deal with the extension of Tomkinson’s definition of the centrally restricted product of finite groups w17, p. 29x to the centrally complete product of the Cherniko¨ groups Gi : i g I 4 , denoted by Zr*Gi , which is the subgroup of the cartesian product consisting of all elements with only finitely many non-central components. Its torsion subgroup is the centrally restricted product, denoted by ZrGi . Also, we put ZrhU Gi s ŽŁ Zi .ŽŁ h Gi . and Zrh Gi the torsion subgroup of ZrhU Gi . We shall be considering groups which are subgroups of Zr*Gi and ZrGi with Gi Chernikov and, as in w5x, we use the notation G g Zr*C and G g ZrC to indicate that G can be embedded in this way.
2. PRELIMINARY RESULTS Although we have not been able to extend Gorchakov’s theorem as completely as we wish, we have been able to establish a version of that and Tomkinson’s theorem for CC-groups having a Chernikov residual system which enjoys certain countability properties related to subgroups of the form X G , where X is a finite subset of G.
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We say that a group G is a c-residually Cherniko¨ group if it has a Chernikov residual system Ni : i g I 4 such that, for every finite subset X of G, X G l Ni : i g I 4 is a countable set. In this case Ni : i g I 4 is said to be a Ž Cherniko¨ . c-system of G. Clearly, a c-residually Chernikov group is residually Chernikov and we leave as an open question whether every residually Chernikov CC-group has a c-system. LEMMA 1. group G.
Let Ni : i g I 4 be a c-system of the residually Cherniko¨
Ž1. If H F G, then H l Ni : i g I 4 is a c-system of H. Ž2. If J : I and N s F Nj : j g J 4 , then NjrN : j g J 4 is a c-system of GrN. Proof. It is an immediate consequence of the definitions. We recall that, in general, a class of groups with the form RX is not Q-closed. This is the reason why we study properties such as Ž2. above, whose practical use is the following: Let Ni : i g I 4 be a c-system of the group G and Gk s GrNk , for every k g I, so that G F ŁGk . Then GŽ J . admits the family GŽ J . l ŽŁ Gk : k g I, k / j4 : j g J 4 as a c-system, where J : I. The next results are very important for our purposes of classification. They provide some theoretical examples of c-residually Chernikov groups. THEOREM 1. For an arbitrary CC-group G, the factor group GrZŽ G . is always a c-residually Cherniko¨ group. Proof. If Z s ZŽ G ., we shall show that CG Ž g G .rZ : g g G4 is a c-system of GrZ. We clearly may write the normal closure of any finite subset of GrZ in the form X G ZrZ, where X is a finite subset of G. Call K s X G , for a given X. If g g G, then it is clear that we have KZrZ l CG Ž g G .rZ s CK Ž g G . ZrZ, so it is enough to show that CK Ž g G . : g g G4 is a countable set for every K, as above. The factor group GrCG Ž K . is Chernikov and so it is countable. Let Y be a countable transversal of G modulo CG Ž K .. If g g G, we may write g s yz, where y g Y and z g CG Ž K .. Then we easily check that CK Ž g G . s CK Ž y G ., and thus the sets CK Ž g G . : g g G4 and CK Ž y G . : y g Y 4 have the same cardinality, so they are countable. A Prufer p-group is an example of a residually Chernikov FC-group ¨ which is not a residually finite group. Thus, the class of residually Chernikov FC-groups lies properly between those of residually finite FC-groups and of residually Chernikov CC-groups but groups of this type do have a c-system, as our next result shows.
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A residually Cherniko¨ FC-group is a c-residually Cherniko¨
Proof. This is clear since if X is a finite subset of the FC-group G, then X G is a finitely generated FC-group and so has only countably many subgroups Žsee w10x..
3. EXTENSION OF GORCHAKOV’S THEOREM TO CC-GROUPS Before dealing with the extension of Gorchakov’s theorem, we need some auxiliary results, which we shall show now. LEMMA 2. Let H be a subgroup of the Cherniko¨ group G. If R is the radicable part of G Ž see w11x, then R s w R, H xCR Ž H .. Proof. Since the subgroup H l R contains the radicable part of H, it has finite index in H. Then H s Ž H l R . F, where F is finite. By w15, Lemma 3.29.1x, R s w R, F xCR Ž F .. Note that R is abelian, so w R, F x s w R, H x, CR Ž F . s CR Ž H .. LEMMA 3. Let Fi : i g I 4 be a family of isomorphic Cherniko¨ groups and let h be an infinite cardinal such that h F < I <. If Z s Ł ZŽ Fi . and D s Ł h Fi , then D is a direct factor of ZD, that is, ZD s D = A, where A is an abelian group. Consequently Zr*Fi g SC / 1C, if h s / 0 , and ZrhU Fi g SCC, if h G / 1. Proof. For each i g I, we put ZŽ Fi . s A i = Bi , where A i is a finite group Bi is a radicable Chernikov group, that is, an abelian Chernikov group. Let A s Ł A i and B s Ł Bi . It is clear that Ž A l D . l A n s A n l D s Ž A l D . n for every n G 1, so A l D is a pure subgroup of A. Since A l D has a finite exponent, it is a direct factor of A, that is, A s Ž A l D . = E Žsee w2, Theorem 27.5x.. Suppose that b s ad, where a g A, b g B, and d g D. Thus
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The proof of Gorchakov’s theorem that we carry out below is based on the original as given in w17x. Despite this, our proof has required several changes; for example, the role played by the centre ZŽ G . in the FC-case is taken by a larger subgroup here. This further changes in the computations. THEOREM 3 ŽGorchakov’s theorem.. Let Ni : i g I 4 be a c-system of the CC-group G such that GrNi ( GrNj , for any pair of indices i, j g I. Then G g SC / I C. Proof. Put Fi s GrNi , for every i g I, so that we may assume that G F Ł Fi , where the Fi are isomorphic Chernikov groups. We shall show the result by induction as follows. Since each Fi g MIN, we may assume that whenever H F Ł Ej : j g J 4 is a CC-group having a c-system with the form H l Ker p j : j g J 4 , where J : I, the Ej are isomorphic groups, and each Ej has an isomorphic copy properly contained in Fj , then H g SC / 1C. To do that, we shall split the proof into several steps. For each i g I, let R i s RŽ Fi . be the radicable part of Fi . We consider R s Ž R i ZŽ Fi . : i g I 4 . R is abelian and GrR is an FC-group, because Fpy1 i RŽ G . F R w11, Lemma 2.1x. Put G s GrR and Fis FirR i ZŽ Fi ., i g I. Then GŽ i . s Fi, i g I, G F ŁFi, and, if J : I, GŽ J . ( GŽ J .rRŽ J .. J is called homogeneous if, for every j g J, the map p j: GŽ J . ª Fj is an isomorphism. Clearly, any subset of homogeneous set is also homogeneous. Concerning homogeneous sets, we quote the following result Žsee w17x.. Ža. E¨ ery homogeneous subset of I is contained in a maximal one and two different maximal homogeneous subsets of I are disjoint. By Ža. above and the fact that any atomic subset of I is homogenous, one has that the different maximal homogeneous subsets of I form a partition of I. They are called the homogeneous components for G in I. The next fact actually shows that these components are well behaved under projections w17, Lemma 2.16x. Žb. If Jl : l g L4 are the homogeneous components for G in I and J : I, then the non-empty members of the set Jl l J : l g L4 are the homogeneous components for GŽ J . in J. The next result describes the structure that a homogeneous component determines on its corresponding projection. Žc. Let J be a homogeneous component for G in I. Then we ha¨ e the normal decomposition GŽ J . s HL, that is, H and L are normal subgroups of GŽ J ., where L is an abelian group and H is the normal closure in GŽ J . of a finite set. In particular H is Cherniko¨ -by-Ž finitely generated abelian. and countable. Moreo¨ er there exists a countable subset K of J such that H ( H Ž K ..
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Since J is homogeneous, we have that GŽ J . ( Fj, for every j g J. Each factor FjrR j is finite, and so is GŽ J .. But GŽ J . ( GŽ J .rRŽ J ., so GŽ J . s RŽ J . H, where H is the normal closure in GŽ J . of a finite set and we note that L s RŽ J . is abelian. Therefore the first assertion of Žc. is proved. Moreover, GŽ J . s GrŽF Ker a j : j g J 4., where a j : G ª Fj is the canonical map, j g J. Also, if j g J and a j: GŽ J . ª Fj denotes the natural projection, since H l ŽF Ker a j : j g J 4. s 1, for every 1 / h g H, there is j s jŽ h. g J such that h f Ker a j. If K s jŽ h. : 1 / h g H 4 : J, K is countable and H l ŽF Ker a j : j g K 4. s 1. Thus H ( H Ž K ., which shows Žc.. To apply the induction, we need the following concept. A subset J of I is said to be a reduced subset of I if one can embed the group GŽ J . in a cartesian product of isomorphic Chernikov groups and these groups are isomorphic to proper subgroups of the Fi . If J is the union of the family Jv : v g V 4 of reduced sets, we may write J as a disjoint union of reduced sets and, bearing in mind that GŽ J . is isomorphic to a subgroup of ŁGŽ Jv ., it follows that J is also a reduced set. Therefore I contains a unique maximal reduced set, J say, which is called the reduced component for G in I. In this case, I y J contains no reduced sets for G. If J is the reduced component for G in I, we assume that GŽ J . F Ł Ej : j g J 4 , and Ej - Fj for all j g J. It is known that there exists only a countable number of isomorphism types of Chernikov groups, Cn : n g N4 say ŽWehrfritz w18x.. If n g N, let Jn s j g J : Ej ( Cn4 ; then J s D Jn , GŽ J . F ŁGŽ Jn ., and GŽ Jn . F Ł Ej, n , where Ei, n ( Cn for all j g Jn . By our induction, we may assume that GŽ Jn . g SC / 1C. Then GŽ J . g SC / 1C and it is enough to show that GŽ I y J . g SC / 1C. Therefore, we suppose that I contains no non-empty reduced subsets and we must show that GŽ I . g SC / 1C. By Ža., we write I as a disjoint union of its homogeneous components, I s D Jl : l g L4 say. For each l g L, we consider the normal decomposition GŽ Jl . s Hl Ll , where Hl is a Chernikov-by-Žfinitely generated abelian . normal subgroup, Ll is an abelian normal subgroup, and Hl ( HlŽ Kl . with Kl countable. If l g L, let Ml s F Ker p j : j g Kl4 so that GrMl ( GŽ Kl .. Let Nl be the inverse image in G of Hl under p Jl. Then Ml l Nl s F Ker p j : j g Jl4 and GrŽ Ml l Nl . ( GŽ Jl .. Observe that G can be embedded in Ž GrŽF Ml .. = Ž GrF Nl .., because ŽF Ml . l ŽF Nl . s 1. But GrNl ( GŽ Jl .rHl s Ll HlrHl is abelian, and so is GrŽF Nl .. By w5, Corollary 1x, GrŽF Nl . g SC / 1C. We now consider the group GrŽF Ml . ( GŽ K ., where K s D Kl . It is clearly sufficient to show that GŽ K . g SC / 1C. By Žb., the countable sets Kl are the homogeneous components for GŽ K . in K. Also, a non-empty reduced set for GŽ K . in K is reduced for G in I; so K has no non-empty reduced subsets. If GŽ K . is replaced by G and K by I the proof of the theorem will be completed
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when we establish Žd. If G F Ł Fi : i g I 4 , the homogeneous components of I are countable sets and G contains some element which has an uncountable number of non-central coordinates, then I contains an uncountable reduced subset. Throughout, we keep the above conditions together with the established notation. We split the proof of Žd. into several steps. Step 1. If H 1 G and GrCG Ž H . is a Cherniko¨ group, then H Ž i . ZŽ Fi . is a proper subgroup of Fi for all but countably many indices i g I. Put J s i g I : H Ž i . ZŽ Fi . s Fi 4 , H1 s H Ž J ., G1 s GŽ J ., and Z j s ZŽ Fj ., j g J. Thus H 1 1 G 1 and Fj s H 1Ž j . Z j , j g J. C G 1Ž H 1 . s G 1 l ŽŁCF ŽŽ H1ŽŁ Z j ..Ž j ... s G1 l ŽŁ Z j . and G1rCG Ž H1 . is Chernikov. Put j 1 R1 s G1 l ŽŁ Z j R j ., so that G1rR1 F ŁŽ FjrZ j R j . and then G1rR1 is residually finite. But G1rR1 is also Chernikov, so it is finite and has finitely many subgroups, L1 , . . . , L s say. If u j : G1rR1 ª FjrZ j R j , j g J, is the projection and Jr s j g J: Ker u j s L r 4 , then there are isomorphism maps Ž G1rR1 .Ž Jr . ª FjrZ j R j , j g Jr , and therefore each Jr is a homogeneous set for G1 in J and for G in I. Hence, by hypothesis, Jr is countable. Therefore J is countable. Given J : I, we add to the above the following notation Ždepending on J .: If A F Ł Fj : j g J 4 , we shall write A s AZrZ, where Z s Ł Z j : j g J 4. Step 2. We now suppose that I contains an uncountable subset, J say, such that G Ž J . F ŽŁ / 1 Fj : j g J 4. H, where H is the normal closure in GŽ J . of a finite set and H Ž j . / 1 for e¨ ery j g J. Then J contains an uncountable reduced set for G. Clearly we may assume that J s I. Put L s GŽŁ Zi ., so that Z s Ł Zi F ZŽ L.. In fact, we have Z s ZŽ L., because the corresponding projections are onto maps. Since L s GZ, we have that L s G F PH, where P s Ł / 1 Fi. Thus L s Ž L l P . H. Let E be the inverse image of L l P under the projection of L in L. If g g L, there are x g E and y g H such that the image under the above projection of g Ž xy .y1 is trivial, so g Ž xy .y1 g Z. Thus g s xyz s Ž xz . y, where z g Z, and we note that xz g E. Then L F EH, and so L s EH. Similarly, one can show that E s DZ, where D s L l ŽŁ / 1 Fi . s E l ŽŁ / 1 Fi .. Now GrCG Ž H . is a Chernikov group and, by Step 1, K 1 s i g I: H Ž i . Zi s Fi 4 is countable. Moreover L s EH s DZH and we find that D l H F Ł / 1 Fi is countable. Therefore, it follows that K 2 s suppŽ D l H . is countable. Put K s I y Ž K 1 j K 2 .. Then LŽ K . s D Ž K . H Ž K . ZŽ K . and D Ž K . s LŽ K . l ŽŁ / 1 Fk : k g K 4.. DŽ K . l H Ž K . s 1 and, since K is uncountable, we may again assume that I s K. In this case, we may write L s Ž D = H . Z and, if i g I, Zi - H Ž i . Zi and H Ž i . Zi - Fi .
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Put X s ZŽ H . and Y s ZŽ D . so that Ž D = H . l Z s Y = X. Write Z s ŁŽ A i = Bi ., where the A i are isomorphic finite groups and the Bi are isomorphic radicable Chernikov groups. Y s Ł / 1 Zi s Ł / 1 Ž A i = Bi . and, if A s Ł A i , B s Ł Bi , A1 s Ł / 1 A i and B1 s Ł / 1 Bi , Z s A = B, and Y s A1 = B1. We claim that there is a direct decomposition Z s Y = V = U, such that V is countable and X F Y = V s T. For, as in Lemma 3, A s A1 = A 2 and B s B1 = B2 . A 2 has a finite exponent and so it is a direct product of finite cyclic groups, A 2 s Dr C j : j g L4 say. As B2 is divisible, it is a direct product of some Prufer ¨ groups and some copies of the full rational group, B2 s Dr Di : k g K 4 say. Note that the groups C j and D k are countable. Since X is a countable subgroup of Z, there are countable subsets L9 of L and K 9 of K such that X F A1 = Ž Dr C j : j g L9.4 = B2 = Ž Dr Di : k g K 94.. If we put AX2 s Dr C j : j g L94 and B2X s Dr Di : k g K 94 such that A 2 s AX2 = AY2 and B2 s B2X = B2Y , then if V s AX2 = B2X and U s AY2 = B2Y , our claim has been carried out. Put W s XV, T s YW, and Z s YW = U. Thus L s DHZ s DSU, where S s HW, which is countable. Since w D, S x F w D, HZ x s w D, H x s 1, it follows that ZŽ DS . s ZŽ D . ZŽ S . s YXW, and so DS l U s YXV l U F T l U s 1 and L s DS = U. Put I1 s I y suppŽ D l S ., and let d g D, s g S, and u g U be elements such that dsuŽ I1 . s 1. Since suppŽ D l S . is countable, it follows that dsu g L l Ł / 1 Fi s D. Therefore, u g DS l U s 1, ds g D, s g D l S and it readily follows that sŽ I1 . s 1 and dŽ I1 . s 1. This shows that LŽ I1 . s DŽ I1 . = SŽ I1 . = UŽ I1 .. For every i g I1 , we put SŽ i . s H Ž i .W Ž i . F H Ž i . Zi . By hypothesis, H Ž i . g Zi . Since w DŽ i ., SŽ i .x s 1, DŽ i . - Fi . Since U is a subgroup of Z, we similarly have that UŽ i . - Fi . Thus I1 is the required reduced set, and Step 2 is completed. Step 3. The end of the proof of Žd. and of Theorem 3. It suffices to construct a J : I satisfying the conditions of Step 2. Suppose we may pick an x g G such that x Ž i . f Zi for an uncountable number of indices. Replacing this set by I, we may assume that x Ž i . f Zi for all i g I. If we Ž R i . : i g I 4 , then GrS F ŁŽ FirR i . where each factor define S s lpy1 i FirR i is finite with given order, n say. A subset T : G is said to be a parallel set for a non-empty subset J of I if, whenever g / h g T, in the above embedding, one has that Ž gS .Ž j . / Ž hS .Ž j . for every j g J. We clearly have that, if T is a parallel set of some J, then < T < F < FirR i < s n. Thus, by the choice of x, the set x, 14 is parallel for I, so, by the finiteness of n, we may choose a maximal parallel set T0 for I. If K is an uncountable subset of I, then T0 is a parallel set for K, and so is contained in a maximal parallel set for K. Choose an uncountable subset I0 of I and a maximal parallel set T for I0 such that T has largest possible order. Then T is a maximal parallel set for all uncountable subsets of I0 .
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Let T s x 1 , . . . , x t 4 . Since we can multiply the elements of T on the left by any element of G, we may assume that x f T. Now let H s ² x, x 1 , . . . , x t :T . Since G is a c-residually Chernikov group, H l Ker p i : i g I 4 is countable, and then there is J : I uncountable such that H l Ker p i s H l Ker p j for every pair i, j g J. Replacing J by I, we may assume that whenever h g H and hŽ i . s 1 for some i g I, then hŽ i . s 1 for every i g I, that is, h s 1. If g g G, then, for some 1 F j F t and .Ž . i g I, Ž gS .Ž i . s Ž x j S .Ž i ., or Ž xy1 j g i s r i g R i . By Lemma 2, R i s CR iŽ H Ž i ..w R i , H Ž i .x, so ri s c i h i , where c i g CR iŽ H Ž i .. and h i g w R i , H Ž i .x. Since w R i , H Ž i .x F H Ž i ., it follows that there is some h g H such that y1 .Ž . y1 xŽ . hŽ i . s h i , and then Ž xy1 i s c i . Therefore, w h, xy1 i s 1 and j gh j gh y1 y1 x w Ž . so H, x j gh s 1. That is, g g HCG H and G s HCG Ž H .. Thus CG Ž H .Ž i . - Fi , since x g H and x Ž i . f Zi , i g I. If CG Ž H . F Zr/ 1 Fi s ŽŁ Zi .ŽŁ / 1 Fi ., proceeding as in Step 2 and taking J s I, we show that G F ŽŁ / 1 Fi : i g I 4. H and we see that all the conditions are satisfied. Then the proof of Žd. is finished. We may now deduce from these facts the proof of Gorchakov’s theorem. If we assume that C s CG Ž H . g Zr/ 1 Fi , there is some c g C with an uncountable number of non-central coordinates. Put Ci s C Ž i . - Fi , i g I. Since the number of isomorphism types of Chernikov groups is countable, there is an uncountable subset I1 of I such that cŽ i . is not central and the Ci are isomorphic, i g I1. By Lemma 1 and by induction, there is an uncountable J : I such that C Ž J . F ŽŁ / 1 C j : j g J 4.H1 and the conditions of Step 2 are satisfied. But ZŽ C . s C l ZŽ G ., and, in particular C js C jrZŽ C j . s Ž CrZŽ C ..Ž j . F Ž GrZŽ G ..Ž j ., for every j g J. Thus G Ž J . F ŽŽŁ / 1 C j : j g J 4. H 1 . H Ž J . F ŽŁ / 1 Fj : j g J 4. ŽH1H .Ž J .. Since ŽH1H .Ž J . is the normal closure of some finite set and ŽH1H .Ž j . / 1, j g J, by Step 2, the proof is completed.
4. EXTENSION OF TOMKINSON’S THEOREM TO CC-GROUPS AND SOME OF ITS APPLICATIONS As an important consequence of Theorem 3, we may classify the c-residually Chernikov CC-groups as subgroups of centrally complete products of Chernikov groups, which is an extension of the CC-case of Tomkinson’s theorem w16x. THEOREM 4. For a group G, the following conditions are equi¨ alent. Ž1. G is a c-residually Cherniko¨ CC-group. Ž2. G g Zr*C. Further, if G is periodic, Ž2. can be replaced by Ž29. G g ZrC.
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Proof. Clearly, the second assertion is a consequence of the first one together with the obvious fact that a periodic Zr*C-group is a ZrC-group. Ž2. « Ž1. Let Gi : i g I 4 be a family of Chernikov groups. The classes of CC-groups and c-residually Chernikov groups are S-closed Žsee Lemma 1., so it suffices to show that the whole group G s Zr*Gi satisfies Ž1.. Therefore it is enough to prove that G in fact has a c-system. Let Cn : n g N4 be a representative set of isomorphism types of Chernikov groups. For every n G 1, put In s i g I : Gi ( Cn4 . By Lemma 3, we have that Zr* Gi : i g In4 g SC / 1C and, since I is the union of the In , G g SC / 1C also. Suppose that G F Ł / 1 Hj : j g J 4 , where the Hj are Chernikov. If p j : G ª Hj is the canonical projection and X : G is a finite set, suppŽ X G . is countable, while if j / suppŽ X G ., then X G F G l Ker p j . Therefore X G l Ker p j : j g J 4 is countable, and so it is in fact a c-system as desired. Ž1. « Ž2. Suppose first that G F Ł Gi : i g I 4 is a CC-group, where the Gi are Chernikov groups, each map p i : G ª Gi is onto, i g I, and G l Ker p i : i g I 4 is a c-system of G. As above, let Cn : n g N4 be a representative set of isomorphism types of Chernikov groups. For each n G 1, let In s i g I: Gi ( Cn4 . Then I is the union of the In , and G F GŽ I1 . = GŽ I2 . = . . . ,GŽ In . F Ł Gi : i g In4 , n G 1. By Lemma 1, GŽ In . is a CC-group admitting GŽ In . l Ker p i : i g In4 as a c-system. Then, by Theorem 3, GŽ In . g SC / 1C, and so G g SC / 1C. Thus, changing the notation if necessary, we may assume that G F Ł / 1 Gi : i g I 4 and show our result by induction on < I <. If I is countable, it follows from w4, Theorem 3.1x, so we may assume that I s a : a - r 4 is uncountable, where r is the least ordinal such that < r < s < I <; clearly, r has to be a limit ordinal. We shall show that I can be expressed as a disjoint union of certain subsets Ja , a - r , such that, if Ia s D Jb : b - a 4 , then the following conditions are satisfied. Ža. a g Ia j Ja . Žb. < Ia j Ja < F max / 0 , < a <4 . Žc. If we define Ha s G l ŽŁ Gg : g g Ia 4 , Ma s G l ŽŁ Gg : g f Ia 4 and Ca s CG Ž CrMa ., then one has that G s Haq1Ca , for every a - r . Given a - r , we assume defined every Jb , b - a , satisfying the conditions: Ža9. d g Id j Jd , for every d - a . Žb9. < Id j Jd < F max / 0 , < d <4 , for every d - a . Žc9. G s Hdq1Cd , for every d such that d q 1 - a . If a is a limit, we define Ja s a 4 , if a f Ia , and Ja s B, otherwise. Clearly each Ja is disjoint with the previous Jb , in such a way that the
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conditions Ža9. and Žb9. are satisfied for d s a , and the condition Žc9. is satisfied, for every d - a . If a s « q 1, then GrM« embeds in P Gd : d g I« 4 and, by w12, Lemma 6.3x, < GrC« < F max a 1 , < « <4 . Therefore, there is an S« F G such that G s S« C« and < S« < F max a 0 , < « <4 . Let K« be the set of all indices g such that the group S« admits a non-trivial projection on the component Gg . But, each element has countable support, so that < K« < F max a 0 , < « <4 . If Ja s Ž a 4 j K« . y Ia , then Ja and the Jb are disjoint and the conditions Ža9. and Žb9. hold for d s a . Further, G s Ha C« , so Žc9. holds for all d - a , and we may define Ja satisfying Ža9., Žb9., and Žc9. and hence Ža., Žb., and Žc.. If Na s G l ŽŁ Gb : b f Jaq1 4. and Da s CG Ž GrNa ., Maq1 F Na , G s Haq2 Caq1 F Haq2 Da , for all a . By w17, Lemma 2.20x, G F Zr*Ž GrNa .. Each GrNa embeds in Ł / 1 Gb : b g Jaq1 4 , so GrNa g Zr*C. Then, by w17, Lemma 2.23x, G g Zr*C. We deduce some consequences of Theorem 4 in order to classify some classes of groups. The next result improves w4, Corollary 5x Žsee also w5, Corollary 2x.. COROLLARY 1. For a CC-group G, we ha¨ e that GrZŽ G . g Zr*C. Furthermore, GrZŽ G . g ZrC pro¨ ided GrZŽ G . is periodic. Proof. It suffices to apply Theorems 1 and 4. We can now extend results first shown by Tomkinson w16x and Gorchakov w7x, in classifying residually Chernikov FC-groups. COROLLARY 2. For a group G, the following conditions are equi¨ alent. Ž1. G is a residually Cherniko¨ FC-group. Ž2. G is isomorphic to a subgroup of a centrally complete product of Cherniko¨ groups, which are central-by-finite groups. Further, if G is periodic, the embedding in Ž2. can in fact be into a centrally restricted product. Proof. As usual, the second assertion is an obvious consequence of the first one. Ž1. « Ž2. By Theorems 2 and 4, G g Zr*C. Also, we may assume that the corresponding components of G are Chernikov FC-groups. Therefore, they are also central-by-finite groups, claimed. Ž2. « Ž1. This is clear. COROLLARY 3. A residually Cherniko¨ FC-group is a QSDŽF j A 0 .group. It is a QSDF-group in the periodic case. Proof. Applying Corollary 2, we see that if G F Zr* Gi : i g I 4 , where the Gi are central-by-finite Chernikov groups, then g g QSDŽF j A 0 .. In fact, since the class QSDŽF j A 0 . is S-closed, we may assume that G s Zr*Gi s ZD, where Z s Ł ZŽ Gi . and D s DrGi , and show that G g
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QSDŽF j A 0 .. But an abelian group can be embedded in the direct product of a periodic abelian group and a torsion-free abelian group and, by w17, Corollary 3.4x, a periodic abelian group is a QSDF-group, so that Z g QSDŽF j A 0 .. If H is a central-by-finite Chernikov group, H s ZŽ H . F, with F finite, and, by w5, Lemma 1x, H g QSDŽF.. Then, since QSDŽF. is D-closed, D g QSDŽF.. Then, by w5, Lemma 1x, G g QSDŽF l A 0 .. The second assertion can be obtained by reasoning in a similar way since in this case Z is a periodic abelian group and so is a QSDF-group. Other special cases arise as nice particular consequences and as an example we develop some of them. The first needs an auxiliary result, which shows a property of countability that does work not in general. Although a Chernikov group is countable, it can contain an uncountable number of subgroups. Let A s ² a i : i G 1: and B s ² bi : i G 1: be two copies of a Prufer ¨ p-group and let G s A = B be their direct product. For each sequence S s Ž « 1 , « 2 , . . . . consisting of ‘‘zeros’’ and ‘‘ones’’, the subgroup of G given as G S s ² a1Ž b 1 . « 1 , a 2 Ž b 2 . « 1 Ž b 1 . « 2 , a3 Ž b 3 . « 1 Ž b 2 . « 2 Ž b1 . « 3 , . . . : is also a Prufer ¨ group and different sequences generate different subgroups of G, so that G has an uncountable number of subgroups. The above G is a radicable group and it is the direct product of two copies of the same Prufer ¨ group. However, if the radicable part of the Chernikov group G is the direct product of Žfinitely many. different Prufer ¨ groups, the situation is very different, as the next result shows. LEMMA 4. We suppose that the radicable part A of the Cherniko¨ group G is a locally cyclic group. Then G only has countably many subgroups. Proof. We may certainly assume that G is an infinite group. Since A is the unique maximal radicable subgroup of G, a subgroup H of G has the form H s BF, where B is a subgroup of A and F is a finite subgroup of G. Since G is countable, it has countably many finite subgroups F only because these are the finite parts of a countably infinite set. Therefore, it will be enough to see that A has only countable many subgroups. But A has the form A s A1 = ??? = A r , where A i is a group of type p`i and the pi are different primes. Thus, if B F A, since the A i have their elements with coprime order, we have that B s Ž B l A1 . = ??? = Ž B l A r .. As each A i has countably many subgroups, the number of subgroups of A is countable, as claimed. COROLLARY 4. Let G be a residually Cherniko¨ CC-group whose radicable part R has rank 1. Then G g Zr*C. Further G & ZrC, pro¨ ided G is periodic.
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Proof. If X : G is finite and T s T Ž X G ., then, by w1, Lemma 1x, T is Chernikov and X GrT is free abelian of finite rank. Thus, if H F X G , HrŽ H l T . is finitely generated, and hence H s Ž H l T . F, where F is finitely generated. X G is countable, and then it contains only a countable number of these F. By Lemma 4, T has a countable number of subgroups, and so has H. It follows immediately that G is a c-residually Chernikov CC-group, and so Theorem 4 applies. COROLLARY 5. Let G be a CC-group such that GrG9ZŽ G . has a finite exponent. Then GrZŽ G . g SDC. Proof. Since G9ZŽ G .rZŽ G . s Ž GrZŽ G ..9, it must be periodic. Also, GrG9ZŽ G . is periodic, and so is GrZŽ G .. By Theorems 1 and 4, GrZŽ G . g ZrC. Put L s GrZŽ G . and assume that L F ZrGi s ZD, where the Gi are Chernikov, Z s T ŽŁ ZŽ Gi .., and D s DrGi . If n G 1, let z s Ž z i . g Z l LD be such that z n g Z l D. Since suppŽ z n . s I0 is finite, if g s Ž g i ., where g i s z i , if i g I0 and g i s 1, otherwise, g g Z l D and z n s g n. Then Z l D is a pure subgroup of Z l LD. But Ž Z l LD .rŽ Z l D . ( Ž Z l LD . DrD s LDrD ( LrŽ L l D . is abelian and isomorphic to a quotient of LrL9 ( GrG9Z. Therefore Ž Z l LD .rŽ Z l D . has finite exponent and it follows that Z l D is a direct summand of Z l LD w2, Theorems 17.2, 28.2x, that is, Z l LD s Ž Z l D . = E, for a certain subgroup E. Therefore LD s Ž Z l LD . D s DE. But D l E s D l E l Z s 1, so we actually have that LD s D = E. By w5, Theorem 2x, E g SDC. Since D g SDC, we have that L g SDC. It is worth mentioning that there is an infinite extraspecial group G which is not a QSDC-group Žsee w17, Example 3.8x.. This group satisfies the hypothesis of Corollary 5 but not of w4, Theorem 5.3x. Therefore this corollary is not a consequence of the result given there.
APPENDIX It is an open question whether the analogue of Tomkinson’s theorem holds in the CC case, that is, whether or not a residually Chernikov CC-group is always a Zr*C-group or, equivalently, if those groups have a c-system. The aim of this appendix is to clarify the status of this question. In generalizing the result due to Gorchakov w7x, we showed in w4, Theorem 4.2x that a residually Chernikov CC-group is a QSD ŽC j A 0 .-group. In fact, Zr*C : QSDŽC j A 0 . w5, Corollary 2x, but this inclusion must be strict, since there are quotients of residually Chernikov groups that have no Chernikov residual systems. This example tells us that we cannot expect a full characterization of the latter groups to be given by this result.
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We may ask if there exists an embedding of any residually Chernikov CC-group in a centrally complete product of simpler groups, although they are not necessarily Chernikov groups. The answer to this question is affirmative, when we consider, for example, the countable CC-groups, which were classified in full in w3x. In fact if we make use of the continuum hypothesis, we may prove results such as the following. THEOREM 5. E¨ ery residually Cherniko¨ CC-group G embeds in a centrally complete product of countable CC-groups. Actually, this product can be considered centrally restricted, if the gi¨ en group G is periodic. The proof of this result can be carried out proceeding as in other parts of this paper and using other, more specific techniques, such as the continuum hypothesis. For these reasons, we shall give only a brief sketch of it. First, we shall need an auxiliary result. LEMMA 5. If G is a residually Cherniko¨ CC-group, then G embeds in a cartesian product of Cherniko¨ groups with
ACKNOWLEDGMENT Both authors are very grateful to Dr. Mike J. Tomkinson for his valuable suggestions and his hospitality at the Glasgow University, where many ideas of this work were carried out.
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REFERENCES 1. J. Alcazar and J. Otal, Slylow subgroups of groups with Chernikov conjugacy classes, J. Algebra 110 Ž1987., 507]513. 2. L. Fuchs, ‘‘Infinite Abelian Groups,’’ Academic Press, New York, 1970]1973. 3. M. Gonzalez and J. Otal, P. Hall’s covering group and embedding of countable CC-groups, Comm. Algebra 18 Ž1990., 3405]3412. 4. M. Gonzalez and J. Otal, Embedding theorems for residually Chernikov CC-groups, Proc. Amer. Math. Soc. 123 Ž1995., 2323]2332. 5. M. Gonzalez and J. Otal, Embeddings of central-by-Chernikov groups and central products of Chernikov groups, preprint. 6. M. Gonzalez and J. Otal, Embedding of CC-groups: A survey, preprint. 7. Yu. M. Gorchakov, Locally normal groups, Siberian Math. J. 12 Ž1971., 907]916. 8. Yu. M. Gorchakov, Theorems of Prufer]Kulikov type, Algebra and Logic 13 Ž1974., ¨ 374]378. 9. P. Hall, Periodic FC-groups, J. London Math. Soc. 34 Ž1959., 289]304. 10. B. H. Neumann, Groups with finite classes of conjugate elements, Proc. London Math. Soc. Ž 3 . 1 Ž1951., 178]187. 11. J. Otal and J. M. Pena, ˜ Characterizations of the conjugacy of Sylov p-subgroups of CC-groups. Proc. Amer. Math. Soc. 106 Ž1989., 605]610. 12. J. Otal, J. M. Pena, ˜ and M. J. Tomkinson, Locally inner automorphisms of CC-groups, J. Algebra 141 Ž1991., 382]398. 13. Ya. D. Polovickiı, ˘ On locally extremal groups and groups with the condition of pminimality, So¨ iet Math. Dokl. 2 Ž1961., 780]782. 14. Ya. D. Polovickiı, ˘ Groups with extremal classes of conjugate elements, Sibirsk. Mat. Zh. 5 Ž1974., 891]895. 15. D. J. S. Robinson, ‘‘Finiteness Conditions and Generalized Soluble Groups,’’ SpringerVerlag, Berlin 1972. 16. M. J. Tomkinson, A characterization of residually finite periodic FC-groups, Bull. London Math. Soc. 13 Ž1981., 133]137. 17. M. J. Tomkinson, ‘‘FC-Groups,’’ Pitman, London, 1984. 18. B. A. F. Wehrfritz, Sylov subgroups of locally finite groups with Min-p, J. London Math. Soc. (2) 1 Ž1969., 421]427.