On Compact Just-Non-Lie Groups

Transcription

1 On Compact Just-Non-Lie Groups FRANCESCO RUSSO Mathematics Department of Naples, Naples, Italy Abstract. A compact group is called a compact Just-Non-Lie group or a compact JNL group if it is not a Lie group but all of its proper Hausdorff quotient groups are Lie groups. We provide information on the structure of these groups. The main results are essentially two. The first result states that a compact JNL group is totally disconnected; the second result describes some circumstances in which a compact JNL group splits over a suitable central subgroup. Mathematics Subject Classification: 22C05; 20E22; 20E34. Key Words and Phrases: compact Just-Non-Lie groups, centerfree compact groups. 1. Introduction. Let X be a class of groups. A group G which belongs to X is said to be an X-group. A group G is said to be a Just-Non-X group, or a JNX group, if it is not an X-group but all of its proper quotients are X-groups. Of course, every simple group which is not an X-group is a Just-Non-X group, so that in the investigation concerning Just-Non-X groups it is natural to require that they have nontrivial Fitting subgroup, that is, they contain a nontrivial abelian normal subgroup. The structure of Just-Non-X groups has already been studied for several choices of the class X, so there is a well developed theory about this topic (see [4]). Moreover the study of Just-Non-X groups has been investigated both in theory of finite groups and theory of infinite groups and many techniques have general application. A reason of interest in Just-Non-X groups is due to a work of M.F.Newman (see [4, Chapter 11] and [4, Theorems 11.1, 11.2]). He proved that a Just- Non-Abelian group, that is, a group which is not abelian but all of its proper quotients are abelian, is characterized to be a homomorphic image of a direct product of an extra-special p-group (p is a prime) by a locally cyclic p-group. This result is relevant because it implies that a Just-Non-Abelian group splits 1

2 on its Fitting subgroup. If X is a class of groups wider than the class of abelian groups, then there are weak formulations of the above result (see [4]). Also here the Fitting subgroup plays a crucial role in the structure of a Just-Non-X group. There are some cautionary observations which are necessary to note, in order to have an approach as in [4] to the class of Lie groups. The existence of a topology in a group does not allow us to speak in an usual way either of formations or of varieties of groups. Literature on varieties of topological groups is recent, as shown in [5], [6], [7], [8], [9], [10]. In particular most of the classical results of [4, Chapter 2] do not apply to topological groups. In the context of topological groups it is therefore not even clear how to define the Fitting subgroup of a topological Just-Non-X group. When we consider the property to be a Lie group, we are moving in the category of Hausdorff topological groups (see [1, p.294]) with corresponding morphisms. In order to speak about quotients in a meaningful way in this category, we should refer to quotients modulo closed normal subgroups (see [1, Definition 1.9]). Many situations in this category show that we may not have any closed normal subgroup at all. Then it would seem reasonable to pick a category consisting of Hausdorff topological groups which have enough Lie group quotients to separate the points, in particular the category of compact groups satisfies such requirements. They are well-known and summarized in [1]. A compact group G is called a compact Lie group if it has a faithful finite dimensional representation (see [1, Definition 2.41]). If N is a normal closed subgroup of a compact group G, it is possible to consider the homogeneous space G modulo N which is a compact group with the quotient topology induced by N. We will refer always to this sense of quotients in the present paper. If G is a topological group, let N (G) denote the set of all normal subgroups of G such that N N (G) if and only if G/N is a Lie group. Then G N (G); further {1} N (G) if and only if G is a Lie group. If N N (G) and M is a closed normal subgroup of G such that N M, then M N (G). If G is a compact group, then N (G) converges to 1 and the natural morphism G lim N N (G) G/N is an isomorphism of compact groups (see [1, p.17-23]). The connected component of the identity will be denoted G 0 (see [1, p.23]). A group G is said to be a compact Just-Non-Lie group or a compact JNL group if G is a compact non-lie group such that all closed normal subgroups N {1} are contained in N (G). Following the previous definition of compact JNL group, we may investigate without ambiguity those compact groups which are non-lie groups, but are rich of Lie-quotients. Under this point of view, our aims are close to the aims of [4], then we follow the classical approach of studying groups which have a prescribed property but whose proper quotient groups do not have it. In Section 2 some general properties of compact JNL group are discussed. Section 3 treats situations in which compact JNL groups have a splitting in a semidirect product (see [2] and [4]). However, we shall not arrive at a satisfactory description of arbitrary compact JNL groups; our results concern special circumstances. Most of our notation is standard and has been referred to [1]. For the general 2

3 properties of compact groups we refer to [1], while the strategy which has been used in many proofs can be found in [4]. 2. General Properties of Compact Just-Non-Lie Groups. Let G be a topological group. Following [1], we say that G has no small subgroups, respectively, no small normal subgroups if there is a neighborhood U of the identity such that for every subgroup, respectively, normal subgroup H of G if H is contained in U then H is trivial. A compact Lie group G is characterized to satisfying one of the equivalent conditions of the following lemma. This has been recalled for convenience of the reader. Lemma 2.1. Let G be a compact group and K denote the field of real numbers or the field of complex numbers. The following statements are equivalent: (a) G has a faithful finite dimensional representation. (b) G has a faithful finite dimensional orthogonal (or unitary) representation. (c) G is isomorphic as topological group to a (compact) group of orthogonal (or unitary) matrices. (d) G is isomorphic as topological group to a closed subgroup of the multiplicative group of some Banach algebra A over K. (e) There is a Banach algebra A over K and an injective morphism from G into the multiplicative group A 1 of A. (f) G has no small subgroups. (g) G has no small normal subgroups. Proof. See [1, Corollary 2.40]. A first characterization of compact abelian JNL groups can be obtained thanks to [1, Proposition 2.42]. In the next statement we will introduce the character Ĝ of a group G (see [1]). Proposition 2.2. Let G be a compact abelian group. G is a compact JNL group if and only if Ĝ is not a finitely generated abelian group but each Ĝ/N is finitely generated abelian, where N {1} is a closed subgroup of G. Proof. A compact abelian Lie group is characterized by [1, Proposition 2.42] only as that group which is a character group of a finitely generated abelian group. This means that a compact abelian group G is a compact Lie group if and only if Ĝ is an abelian group which is finitely generated. By negation, G is not a compact Lie group if and only if Ĝ is not a finitely generated abelian 3

4 group. Since all closed subgroups N {1} are contained in N (G), Ĝ/N has to be finitely generated abelian by [1, Proposition 2.42]. Now the result follows. Example 2.3. Let p be a prime number. The group Z p of p-adic integers (see [1, Example 1.28]) is a torsion-free abelian compact JNL group, as its character group Ẑp = Z(p ) is the Prüfer group. Z p is not a Lie group but the proper quotients of Z p are discrete cyclic of p-power order and these are Lie groups. We note that Z p has a nonsingleton closed normal abelian subgroup A = pz p of index p and Z p does not split over A. This implies that a compact JNL group can not split over a normal closed nonsingleton subgroup whose index is finite. From now the symbol Z p will denote always the group of Example 2.3. In the category of compact groups, we may extend a Lie group by another Lie group and we will obtain again a Lie group. This follows easily from Lemma 2.1. Lemma 2.4. Let G be a compact JNL group. (i) G does not have a closed normal nonsingleton Lie subgroup. (ii) G does not have a finite normal nonsingleton subgroup. (iii) If G is abelian, then G is torsion-free. Proof. (i). If N is a closed normal nonsingleton Lie subgroup, then N N (G) and so G/N is a Lie group. But then G, as an extension of a Lie group by a Lie group is a Lie group, contrary to the definition of compact JNL group. Finally (i) implies (ii) and (ii) implies (iii). Lemma 2.4 and most of the successive statements have been communicated in the form of [3]. Proposition 2.5. Let G be a compact JNL group. If M is a nonsingleton closed normal subgroup of G, then G contains a subgroup which is properly smaller than M. Proof. The set N (G) of all normal subgroups such that G/N is a Lie group is a nontrivial filterbasis intersecting in {1} while not containing {1}. If M is a nonsingleton closed normal subgroup, and G is a compact JNL group, then M N (G). Now there is an N N (G) such that M N. Then M N N (G) is properly smaller than M. Proposition 2.5 implies that a compact JNL group with nonsingleton center contains always a subgroup which is properly smaller than the center. The following notion will be used in the next statement. Let G be a compact group and N be a closed normal subgroup of G. We will say that N has no closed normal subgroups which are core-free in G 4

5 if for each closed normal subgroup M of N, then the subgroup M G {1}, where M G = g G M g is the core of M in G. This terminology is standard for abstract groups and can be found for instance in [12]. Proposition 2.6. Let G be a compact JNL group. (i) If Z(G) 1, then Z(G) is a compact JNL group. Furthermore G is a finite central extension of Z p. (ii) Assume that N {1} is a closed normal subgroup of G. If N has no closed normal subgroups which are core-free in G, then N is a compact JNL group. Proof. (i). By (i) of Lemma 2.4, Z(G) is not a Lie group. Let N {1} be a closed subgroup of Z(G). Then N is a closed normal subgroup of G and thus G/N is a Lie group. In particular, Z(G)/N G/N is a Lie group. We have proved that Z(G) is an abelian JNL group. Since G is profinite, G/Z(G) must be finite. In particular, G/N and Z(G)/N are finite. Therefore Z(G) is isomorphic to Z p so that G is a finite central extension of Z p. (ii). Let M {1} be a closed normal subgroup of N. Then M G {1} and G/M G is a Lie group. Thus the closed subgroup N/M G is a Lie group, too. But then N/M (N/M G )/(M/M G ) is a Lie group as well. Proposition 2.7. Let G be a compact JNL group. (i) If G is abelian, then G Z p for some prime number p. (ii) If G is nilpotent, then G is abelian. Proof. (i). By duality A = Ĝ is an abelian group which is not finitely generated, bu in which every proper subgroup is finitely generated. We write A additively. By (iii) of Lemma 2.4, G is torsion-free and so A is divisible (see [1, Corollary 8.5]). Thus A Q (I) p P Z(p ) (Ip), where I and I p are suitable sets and P denotes the set of all prime numbers (see [1, Theorem A1.42]). Suppose that I. Let a 0, a Q (I). Then 1 2 Z a is a proper subgroup of A that is not finitely generated. This is a contradiction. Thus A is a divisible torsion group. Write A p = Z(p ) (Ip) for its p-primary components. Since A is nonzero, at least one A p is nonzero. Let A (p) denote the sum q p A q of all q-primary components A q for q prime number which is distinct by p. Suppose A (p) {0}. Then A p A and thus A p is finitely generated contradicting the fact that a Prüfer group is not finitely generated. Thus A = Z(p ) (Ip). Let j I p, then A Z(p ) Z(p ) (Ip\{j}). If the second summand were nonzero, then Z(p ) would have to be finitely generated, which is not. Thus I p = {j} and A is a Prüfer group. This causes its dual G to be a group Z p of p-adic integers. 5

6 (ii). If we can prove that G is abelian, whenever G is nilpotent of class at most 2, then we are done, because the second center Z 2 (G) has class of nilpotence at most 2 and would have class 2 if G is nonabelian. Thus we may assume that G is nilpotent of class at most 2 without loss of generality. Then [G, G] Z(G), and [g 1 Z(G), g 2 Z(G)] = {[g 1, g 2 ]}, where g 1, g 2 G, so that the bihomomorphic function b : G G Z(G) factors through a bihomomorphic function B : G/Z(G) G/Z(G) Z(G). Now Z(G) is normal in G and since G {1} we have Z(G) {1} as G is nilpotent. Since G is a compact JNL group, G/Z(G) is a Lie group. Therefore Z(G) is open in G and G/Z(G) is finite. Now [G, G] = B(G/Z(G) G/Z(G)) is a union of compact Lie subgroups g G B(G/Z(G), gz(g)). On the other hand, Z(G) Z p by (i) of Proposition 2.6 and (i) above. In particular, Z(G) does not contain any nonsingleton Lie subgroups. Hence B(G/Z(G), gz(g)) = {1} for each g G and thus [G, G] = 1. Example 2.8. The compact topological ring Z p of p-adic integers contains the multiplicative group E of p 1 roots of unity. So we can form G = Z p E with E acting on Z p by multiplication. Then G is a profinite centerfree metabelian group for each p > 2 (see [11] and [13]). Every nonsingleton normal subgroup of G contains one of the form p k Z p {1} and thus is contained in N (G). Therefore G is a compact JNL group. We note that this example illustrates that solvability of compact JNL groups does not imply commutativity. Proposition 2.7 proves that a compact abelian JNL group is totally disconnected. However, this conclusion holds more generally, as the following result shows. Theorem 2.9. A compact JNL group G is totally disconnected. Proof. Assume that G is a compact JNL group and G 0 {1}. We shall derive a contradiction. (a). Since G is a compact JNL group, G/G 0 is a Lie group and thus, as a totally disconnected group, is finite. (b). We will denote with S the commutator subgroup [G 0, G 0 ] of G 0 and with A the identity component Z(G 0 ) 0 of Z(G 0 ). Both of these subgroups are characteristic subgroups of G 0. Set = S A. We claim that = {1}. Suppose that {1}. Then G/ is a Lie group. In particular, S/ is a Lie group, whence S/Z(S) is a Lie group, since Z(S). The factor group S/Z(S) 6

7 is of the form j J S j for a family of centerfree compact connected simple Lie groups (see [1, Theorem 9.24]), and thus J is finite. Then [1, Theorem 9.19] allows us to conclude that is finite. From (ii) of Lemma 2.4 we have a contradiction. Therefore = {1} and thus we have a direct product decomposition (see [1, Theorem 9.24]). G = S A (c). Suppose that S 1. Then G/S, and therefore G 0 /S, is a Lie group. Hence A G 0 /S is a Lie group. Then (i) of Lemma 2.4 implies that A = {1} and therefore G 0 = S = j J S j. Also, G 0 is centerfree. By Lee s Theorem [1, Theorem 9.41] there is a finite group F such that G = F S. Since the factors S j are simple, the action of F induces a permutation group on J. But F is finite, then there is a finite subset I of J which is invariant under this action. Then j I S j is a nonsingleton normal subgroup of G and is a Lie group as a finite product of Lie groups. Now (i) of Lemma 2.4 implies that S = {1}, and thus we know that G 0 = A is abelian. (d). The factor group Γ = G/A acts as a finite group of automorphisms on A, and every Γ-invariant nonsingleton subgroup B of A is normal in G. Then G/B is a Lie group and thus A/B is a finite dimensional torus (see [1] for details). By Pontryagin duality (see [1, Theorem 1.37]), Γ acts as a finite automorphism group on the character group  and ( ) every proper Γ-invariant subgroup P of  is finitely generated free. We write  additively. Let R = Z[Γ] be the integral group ring: this makes naturally  into an R-module. We claim that rank  <, where rank  denotes the rank of the torsion-free abelian group A (see [1, Appendix 1]). If Γ = n is a positive integer, then rank R = n, and the R-submodule P R generated by an arbitrary subgroup P of  satisfies the condition rank P R n rank P. As a consequence, if we suppose that  has infinite rank, then we can construct a proper R-submodule of  of infinite rank in the following way. We take infinitely many distinct elements â 1, â 2,... in  and consider â 1 â This is against ( ). Thus rank  < and we conclude that  is finitely generated free. Then  is a finite dimensional torus. But, G would be a Lie group and this cannot be. This final contradiction proves the result. 7

8 Since a totally disconnected compact Lie group is finite (see [1, Exercise 2.8, (ii)]), Theorem 2.9 shows that a compact JNL group is not finite but has each proper quotient which is finite. Conversely if G is a totally disconnected compact group, which is not finite but all whose proper quotients are finite, then G is obviously a compact JNL group. Corollary Assume that G is a compact JNL group. If M is a nonsingleton closed normal nilpotent subgroup of G, then M is isomorphic to Z p for a suitable prime p. Proof. Theorem 2.9 implies that G is totally disconnected, then G/M is a finite group. We have also that M/N is finite, where N {1} is a closed normal subgroup of M. If M is a Lie subgroup of G, then G is a Lie group, against the definition of compact JNL group. Then M cannot be a Lie group such that all its closed normal subgroups N {1} are contained in N (M). It follows that M is a compact JNL group. The remainder of the proof follows from Proposition 2.7. In the situation of Corollary 2.10 one can conclude that the smallest closed subgroup F containing all nilpotent normal subgroups of G is abelian. Moreover if F is nonsingleton, then it is isomorphic to some Z p, where p is a suitable prime. This circumstance is analogous to [4, Theorems 10.5, 10.9, 10.10], where the Fitting subgroup is involved. Corollary Assume that G is a compact JNL group and A is a nonsingelton closed abelian normal subgroup of G. If p(a) is a prime which does not divide G/A, then G splits over A, that is, G is the semidirect product of Z p and a finite group which is isomorphic to G/A. Proof. By Theorem 2.9 and Corollary 2.10 there exists a prime p = p(a) such that A is a nonsigleton open abelian normal subgroup of G which is isomorphic to Z p. Since Theorem 2.9 implies that G is totally disconnected, we may apply directly [2, Satz III] and the result follows. In the situation of Corollary 2.11, [2, Satz III] shows that all the complements of A are conjugated via inner automorphisms of G. This circumstance is not new, because it can be found in most of the classical results on Just-Non-X groups, where X is a given class of groups. We are referring to situations as in [4, Theorems 11.1, 11.2, 12.26, 12.30, 14.1, 14.2, 14.8, 14.10, 14.18, 14.19, 15.4, 15.5, 15.11, 16.21, 16.24, 16.28, 16.30, 16.31, 16.32, 16.33, 17.5, 17.7, 17.8, 17.9] and [4, Corollaries 12.27, 12.28, 12.29]. 3. Structure of Certain Compact Just-Non-Lie Groups. 8

9 After the remarks of Section 2, one can see whether similar circumstances can be extended to profinite groups, when one wishes to discuss topological groups which are not Lie groups but all whose proper (topological) quotients are Lie groups. Clearly a profinite group is a compact JNL group if and only if every nonsingleton closed normal subgroup is open. However the consideration of the Nottingham Group can be useful to visualize profinite groups which are neither compact nor Lie groups but all whose proper quotients are Lie groups. Example 3.1. Let p be a prime number. Following [13,p.66-67], F p [t] denotes the formal power series algebra over the field with p elements F p in an indeterminate t. Write A for the group of (continuous) automorphisms of F p [t] and for each integer n 1 let J n be the kernel of the homomorphism from A to the automorphism group of F p [t]/(t n+1 ), where (t n+1 ) is the ideal generated by t n+1. From [13,p.66-67], it is known that J 1 coincides with the inverse limit of J 1 /J n for n 1 and each J 1 /J n is a finite group of order p n. Thus J 1 is a pro-p group. Moreover J 1 is centerfree, profinite, pronilpotent, is not a Lie group, has each nonsingleton closed normal subgroup which is not a Lie group. Here J 1 has each proper quotient which is a Lie group, but J 1 is not a compact group since it is not a strictly projective limit of compact Lie groups [1, Corollary 2.43]. From [13, Propositions 2.2.2, 2.3.2, 2.4.3], the problem to classify all profinite (respectively prosolvable, respectively pronilpotent) groups, which are not Lie groups but all whose proper quotients are Lie groups, can be reduced to the corresponding problem for pro-p groups (p is a prime). However, we appear to know nothing about a complete classification for such groups. Coming back to compact JNL groups, a rich example of solvable compact JNL group which allows us to visualize the situation in the non-abelian case is the following. Example 3.2 (K.H.Hofmann). We take a prime number p 2 and let A = Z 2 p be the free Z p -module of rank 2. Every closed (additive) subgroup of A is obviously a free Z p -module of rank at most 2. A Z p -submodule of rank 2 is an open subgroup of A, and, equivalently, has finite index in A. A Z p -submodule of rank 1 of A is of the form Z p (a, b) for each a, b Z p. Let R Z p \ pz denote the multiplicative group of (p 1)-th roots of unity. Let Γ denote a group of automorphisms of A with the matrix representations ( a 0 0 b ), ( 0 a b 0 where a, b R We note that Γ is a group of monomial matrices and it is isomorphic to a semidirect product of the diagonal subgroup of R 2 by the cyclic group of order 2. In particular, Γ = 2(p 1) 2. Now let G = Γ A ) 9

10 denote the semidirect product with respect to the natural action of Γ on A. We will see that ( ) G is a compact JNL group. Let N be a nonsingleton closed normal subgroup of G. We must show that N has finite index in G. Since it suffices to show that the normal subgroup N (A {1}) has finite index in G, we may assume that N = B {1}, where B is a Γ-invariant Z p -submodule of A. We must show that rank B=2, for then B is open in A; therefore A/B is finite. If rank B=0, then N = {1} and this is not relevant for the definition of compact JNL group. Assume that rank B=1. We will get to a contradiction. Now B = Z p (a, b) for suitable elements a, b, Z p, not both of which are zero. Since B is Γ-invariant, for each γ Γ there is a nonzero λ = λ γ Z p such that γ(a, b) = λ (a, b). If b = 0, then a 0 and we let τ = ( whence (λ τ a, 0) = λ τ (a, 0) = τ(a, 0) = (0, a), which is impossible. Likewise a = 0 is impossible, and so a 0 b. Then (λ τ a, λ τ b) = λ τ (a, b) = τ(a, b) = (b, a), and so λ τ = b/a = a/b. We conclude that (a + b)(a b) = a 2 b 2 = 0, then either a = b or a = b. We set ( ) r 0 α = 0 1 for some 1 r R. The existence of such r is due to the fact that p 2. Then, in the first case, λ α a, λ α a) = λ α (a, b) = α(a, b) = (ra, b). We first conclude λ α = 1, then a = ra and so r = u, a contradiction. In the second case, we obtain λ α a = ra, then λ α a = λ α b = b = a. So again we get λ α = 1 and r = 1. This final contradiction proves ( ). From Proposition 2.7 a compact abelian JNL group is isomorphic to Z p, but in our case A Z p. Then the construction of our group G shows that (i) G is a solvable compact JNL group with a nonsingleton abelian normal closed subgroup A {1} such that A has finite index in G and is not a compact JNL group; (ii) G is a compact JNL group which is profinite and solvable of derived length 3. Moreover G is abelian and G Z p. (iii) G is centerfree. (iv) By taking the direct product of finitely many finite cyclic groups and G, we may construct a centerfree solvable compact JNL group of arbitrary derived length. ), The group Z p has many nonclosed proper nonsingleton subgroups. We might wish to replace E by the full group of units Z p \ pz p of Z p in Example 2.8. The 10

11 result is a more complicated metabelian profinite group, but also one that is not a compact JNL group. Still, observations like the following can be made. Proposition 3.3. Let G be a compact JNL group. There is a descending sequence G = G 1 G 2 G 3... {1} of closed normal subgroups of G converging to 1 such that G n /G n+1 is a finite product of simple groups or groups of prime order, for each positive integer n 1. In particular, G is a second countable and thus metric profinite group. Proof. The totally disconnected compact JNL group G cannot be finite, since it is not a Lie group. Then it has a descending family of compact normal subgroups G = G 1 G 2 G 3... converging to 1, such that each factor group G n /G n+1 is a finite product of simple groups or groups of prime order, for each positive integer n 1 (see [1, Theorem 9.91]). If G n+1, then G n {1}, and thus G n N (G). Hence G/G n is a Lie group and thus is finite since G is totally disconnected. Proposition 3.4. Let G be a compact JNL group and A be a nonsingleton closed central subgroup of G. Then G splits over A. Proof. Let p be a prime. From Proposition 2.6 G is a finite central extension of Z(G) Z p. From (i) of Proposition 2.7 A Z p. We deduce that G is a finite central extension of A. But, A is torsion-free normal and G/A is finite. Then G splits on A as claimed. A simple consideration can be done in order to have a deep knowledge of compact JNL groups without nonsingleton torsion-free normal subgroups. As it is shown in [1, Theorem 9.23], centerfree compact groups can be easily described in terms of Mayer-Vietoris sequences. Proposition 3.5. Let G be a compact JNL group without nonsingleton torsion-free normal subgroups. Then G is centerfree. Proof. Assume that Z(G) {1} and let z Z(G) with z 1. The subgroup z cannot be neither finite from (ii) of Lemma 2.4 nor torsion-free from the hypothesis. Then z = 1 and we obtain a contradiction. The result is proved. We end the present Section with an instructive construction which has been due to D.J.Robinson and J.Wilson. They have introduced (see [4, Propositions 15.7]) a method to classify a Just-Non-Polycyclic group, that is, a group which is not polycyclic but all of its proper quotients are polycyclic. Many authors have 11

12 followed their idea obtaining a satisfactory description of Just-Non-X groups, where X is a given class of groups. Constructing a suitable wreath product W, D.J.Robinson and J.Wilson embed a Just-Non-Polycyclic group G in W (see [4, Lemma 15.6, Propositions 15.7, 15.8, 15.10, Theorem 15.11]). The following example shows that a similar process can not be done for compact JNL groups. As testified in the proofs of [4, Lemma 15.6, Propositions 15.7, 15.8] a similar process can not be followed for Just-Non-X, where X is an arbitrary class of groups. Example 3.6. Let p be an odd prime, Z p be the group of p-adic integers and Z(2) be the cyclic group of order 2. The group G = Z p wrz(2) = HwrK denotes the standard wreath product of H = Z p by K = Z(2) (see [11, p.32-33]). As known, it is possible to write G = K H K, where B = H K is the direct product of K -copies of H. The subgroup B is often called the base of G. Here we have that Z(G) coincides with all the elements of B which have the same components (diagonal subgroup of B) and Z(G) is non-identic. Our group is not nilpotent since the ascending chain of the centers of G has length ω, then G is hypercentral compact group. But G contains a closed normal subgroup of the form Z n p with n = K > 1; we know from Proposition 2.7 that a closed normal abelian subgroup of a compact JNL group is isomorphic to Z p, and Z n p Z m p if and only if m = n. The consideration of G certainly can not produce compact JNL groups so that an extension of [4, Theorem 15.11] can not be obtained for compact JNL groups. References [1] Hofmann, K.H., and S.A.Morris, The Structure of Compact Groups, Walter de Gruyter, 2006, Berlin. [2] Hofmann, K.H., Zerfällung topologischer Gruppen, Mathematik Zeitschrift 84 (1964), [3] Hofmann, K.H., Personal Communication. [4] Kurdachenko, L., Otal, J., and I.Subbotin, Groups with prescribed quotient subgroups and associated module theory, World-Scientific, 2002, Singapore. [5] Neumann, H., Varieties of groups, Springer-Verlag, 1967, Berlin. [6] Morris, S.A., Varieties of topological groups, Bulletin of the Australian Mathematical Society 1 (1969), [7] Morris, S.A., Varieties of topological groups. II, Bulletin of the Australian Mathematical Society 2 (1970), [8] Morris, S.A., Varieties of topological groups. III, Bulletin of the Australian Mathematical Society 2 (1970), [9] Morris, S.A., Varieties of topological groups, Bulletin of the Australian Mathematical Society 3 (1970),

13 [10] Morris, S.A., Varieties of topological groups, a survey, Colloquium Mathematicum vol.xlvi fasc.2 (1982), [11] Ribes, L., and A.Zalesskii, Profinite Groups, Springer-Verlag, 2000, Berlin. [12] Robinson, D.J., A Course in the Theory of Groups, Springer-Verlag, 1980, Berlin. [13] Wilson, J.S., Profinite Groups, Clarendon Press, 1998, Oxford. 13