DikranjanGiordanoAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 10, No. 1, 2009 pp. 85-119 Arnautov’s problems on semitopological isomorphisms Dikran Dikranjan and Anna Giordano Bruno Abstract. Semitopological isomorphisms of topological groups were introduced by Arnautov [2], who posed several questions related to compositions of semitopological isomorphisms and about the groups G (we call them Arnautov groups) such that for every group topology τ on G every semitopological isomorphism with domain (G, τ ) is necessarily open (i.e., a topological isomorphism). We propose a different approach to these problems by introducing appropriate new notions, necessary for a deeper understanding of Arnautov groups. This allows us to find some partial answers and many examples. In particular, we discuss the relation with minimal groups and non-topologizable groups. 2000 AMS Classification: Primary 22A05, 54H11; Secondary: 18A20, 20F38, 20K45. Keywords: A-complete topology, Heisenberg group, Markov group, minimal group, open mapping theorem, permutations group, semitopological isomor- phism, Tăımanov topology, topologizable group. 1. Introduction It is easy to prove that for every continuous isomorphism f : (G, τ ) → (H, σ) of topological groups, there exist a topological group (G̃, τ̃ ) containing G as a topological subgroup and an open continuous homomorphism f̃ : (G̃, τ̃ ) → (H, σ) extending f [2, Theorem 1] (see also [14, Theorem 1.1] for continuous surjective homomorphisms). The following notion is motivated by the fact that it is not always possible to prove the existence of such G̃ and f̃ , asking G to be also a normal subgroup of G̃ (see also [1] for topological rings). 86 D. Dikranjan and A. Giordano Bruno Definition 1.1 ([2, Definition 2]). A continuous isomorphism f : (G, τ ) → (H, σ) of topological groups is semitopological if there exist a topological group (G̃, τ̃ ) containing G as a topological normal subgroup and an open continuous homomorphism f̃ : (G̃, τ̃ ) → (H, σ) extending f . In other words semitopological isomorphisms are restrictions of open contin- uous surjective homomorphisms to normal subgroups. Obviously the class of semitopological isomorphisms contains the class of topological isomorphisms. Arnautov characterized semitopological isomorphisms [2, Theorem 4]. We give his characterization in terms of commutators and of thin subsets, as done in [14]. For a neighborhood U of the neutral element eG of a topological group G call a subset M of G U -thin if ⋂ {x−1U x : x ∈ M} is still a neighborhood of eG (i.e., there exists a neighborhood U1 of eG in G such that xU1x −1 ⊆ U for every x ∈ M ). The subsets M of G that are U -thin for every U are precisely the thin sets in the sense of Tkachenko [29, 30]. For example compact sets are thin. Theorem 1.2 ([2, Theorem 4]). Let (G, τ ) and (H, σ) be topological groups. Let f : (G, τ ) → (H, σ) be a continuous isomorphism. Then f is semitopological if and only if for every U ∈ V(G,τ )(eG): (a) there exists V ∈ V(H,σ)(eH ) such that f −1(V ) is U -thin; (b) for every g ∈ G there exists Vg ∈ V(H,σ)(eH ) such that [g, f −1(Vg)] ⊆ U . In [14] we extended the notion of semitopological isomorphism introducing semitopological homomorphisms. We defined new properties and considered particular cases in order to give internal conditions, similar to those of Theorem 1.2, which are sufficient or necessary for a continuous surjective homomorphism to be semitopological. Finally we established various stability properties of the class of all semitopological homomorphisms. Many particular cases are considered and they turn out to be useful also in this paper as well as other particular results; for those we will give references. In Section 2 we give general properties of semitopological isomorphisms and see some stability properties of the class Si of all semitopological isomorphisms. In fact it has been proved in [2] that the class Si is stable under taking sub- groups, quotients and products, but not under taking compositions. The aim of this paper is to discuss and answer the following problems raised by Arnautov [2]: Problem A ([2, Problem 14]). Find groups G such that for every group topol- ogy τ on G every semitopological isomorphism f : (G, τ ) → (H, σ), where (H, σ) is a topological group, is open. Arnautov’s problems on semitopological isomorphisms 87 Problem B ([2, Problem 13]) Let K be a class of topological groups. Find (G, τ ) ∈ K such that every semitopological isomorphism f : (G, τ ) → (H, σ) in K is open. The third problem concerns compositions: Problem C ([2, Problem 15]) (a) Which are the continuous isomorphisms of topological groups that are compositions of semitopological isomorphisms? (b) Is every continuous isomorphism of topological groups composition of semitopological isomorphisms? 1.1. The Open Mapping Theorem and its weaker versions. According to the Banach’s open mapping theorem every surjective continuous linear map between Banach spaces is open [3]. As a generalization, Pták [22] introduced the notion of B-completeness for the class of linear topological spaces. It was based on the property weaker than openness, that can be formulated also in the larger class of topological groups as follows: a homomorphism f : G → H of topological groups is called almost open, if for every neighborhood U of eG in G the image f (U ) is dense in some neighborhood of eH in H. A topological group G is B-complete (respectively, Br-complete) if every continuous almost open surjective homorphism (respectively, isomorphism) from G to any Hausdorff group is open. These groups were intensively studied in the sixties and the seventies ([4], [15], [16], [27]). It was shown by Husain [16] that locally compact groups as well as complete metrizable groups are B-complete. Brown [4] found a common generalization of this fact by proving that Čech-complete groups are B-complete. The following notion introduced by Choquet (see Döıtchinov [11]) and Ste- phenson [26] in 1970 takes us closer to the spirit of Banach’s open mapping theorem: Definition 1.3. A Hausdorff group topology τ on a group G is minimal if for every continuous isomorphism f : (G, τ ) → H, where H is a Hausdorff topological group, f is a topological isomorphism. Call G totally minimal if for every continuous homomorphism f : (G, τ ) → H, where H is Hausdorff, f is open. Clearly, the totally minimal groups are precisely the topological groups that satisfy the Banach’s open mapping theorem. Since all surjective homomor- phisms between precompact groups are almost open, a precompact group is Br-complete (respectively, B-complete) if and only if it is minimal (respec- tively, totally minimal). In particular, the Br-complete precompact abelian groups coincide with the minimal abelian groups as every minimal abelian group is precompact according to the celebrated Prodanov-Stoyanov’s theo- rem. According to this theorem, an infinite minimal abelian group is never discrete. This radically changes in the non-abelian case. In the forties Markov 88 D. Dikranjan and A. Giordano Bruno asked whether every infinite group G is topologizable (i.e., admits a non-discrete Hausdorff group topology). Definition 1.4. A group G is: • Markov if the discrete topology δG is the unique Hausdorff group topol- ogy on G (i.e., δG is minimal); • totally Markov if G/N is Markov for every N ⊳ G. Obviously totally Markov implies Markov and finite groups are totally Mar- kov, while every simple Markov group is totally Markov. Denote by M and Mt the classes of all Markov and totally Markov groups respectively. Markov’s question (on whether M contains infinite groups), was answered only thirty-five years later by Shelah [24] (who needed CH for his example, resolving simul- taneously also Kurosh’ problem) and Ol′shanskii [21] (who made use of the properties of remarkable Adian’s groups). A smaller class of groups arose in the solution of a specific problem related to categorical compactness in [10]: namely the subclass of Mt consisting of those groups G ∈ Mt such that every subgroup of G belongs to Mt as well (these groups were named hereditarily non-topologizable by Lukács [18]). It is still an open question whether an infinite hereditarily non-topologizable group exists ([9, 10, 18]). A possibility to relax the strong requirement in the open mapping theorem in the definition of minimal groups is to restrict the class of topological groups: Definition 1.5. Let K be a class of topological groups. A topological group (G, τ ) ∈ K is K-minimal if (G, σ) ∈ K and σ ≤ τ imply τ = σ. When K is the class of all metrizable abelian groups, K-minimal groups are precisely the minimal abelian groups that are metrizable [8], but in general a K- minimal group need not be minimal. Anyway, if H is the class of all Hausdorff topological groups, then H-minimality is precisely the usual minimality. Recently new generalizations of minimality for topological groups were con- sidered (relative minimality and co-minimality, cf. [7, 25]). 1.2. Main Results. The next definition reminds the Br-completeness (since we impose openness only on certain continuous isomorphisms, namely, the semi- topological ones): Definition 1.6. A group topology τ on G is A-complete if for every group topology σ on G, σ ≤ τ and idG : (G, τ ) → (G, σ) semitopological imply τ = σ. Finally, we can formulate the notion that captures the core of Problem A: Definition 1.7. A group G is an Arnautov group if every group topology on G is A-complete (i.e., if for every pair of group topologies τ, σ on G with σ < τ , idG : (G, τ ) → (G, σ) is not semitopological). Hence Problem A can be formulate also as follows: characterize the groups G such that every group topology on G is A-complete, that is, characterize the Arnautov groups. Arnautov’s problems on semitopological isomorphisms 89 We denote by A the class of all Arnautov groups. Tăımanov [28] introduced the group topology TG on a group G, which has the family of the centralizers of the elements of G as a prebase of the filter of the neighborhoods of eG. This topology was introduced with the aim of the topologization of abstract groups with Hausdorff group topologies. Since idG : (G, δG) → (G, σ) is semitopological if and only if σ ≥ TG (see [14, Corollary 5.3] or Remark 5.12) and we are studying Arnautov groups, we need to impose that TG is discrete and we introduce the following notion. Definition 1.8. A group G is: • Tăımanov if TG = δG; • totally Tăımanov if G/N is Tăımanov for every N ⊳ G. Obviously every simple Tăımanov group is totally Tăımanov. We denote by T and Tt the classes of Tăımanov and totally Tăımanov groups respectively. Since Problem A in its full generality seems to be hard to handle (because of two universal quantifiers), we start considering a particular case, that is when the discrete topology on a group G is A-complete and we prove that for a group G the discrete topology is A-complete if and only if G ∈ T (see Theorem 5.13). Moreover we extend this result for almost trivial topologies (which are obtained from the trivial ones by extension, as their name suggests — see Section 3), characterizing in Theorem 5.15 when an almost trivial topology is A-complete in terms of T. Moreover Tt contains A, but we do not know if they coincide (see Theorem 5.16 and Question 5.17). Example 5.18 considers properties of the permutations group S(Z) related to Problem A. First of all it shows that A-completeness has a behavior different from that of the usual minimality. Indeed we see that S(Z) admits at least two different but comparable A-complete group topologies. Moreover S(Z) is not Tăımanov and consequently not Arnautov. Nevertheless S(Z)/Sω(Z) is totally Tăımanov but we do not know if it is also Arnautov (see Question 5.20). This question can be seen as a first step in answering the following one, which could give an infinite example of a simple infinite Markov group without assuming CH (see Question 5.27): does S(Z)/Sω(Z) ∈ M? But the situation can be reversed: if S(Z)/Sω(Z) ∈ M then S(Z)/Sω(Z) ∈ A, in view of Corollary 5.26(b), which says that every simple Markov group is necessarily Arnautov. Thanks to this property we have the unique infinite Arnautov group that we know at the moment, that is Shelah group, which is an infinite simple Markov group constructed under CH [24] (see Example 5.29). The next definition, combining Definition 1.6 (A-completeness) and Defini- tion 1.5 (K-minimality) will allow us to handle easier Problem B. 90 D. Dikranjan and A. Giordano Bruno Definition 1.9. For a class K of topological groups, a topological group (G, τ ) from K is AK-complete if (G, σ) ∈ K, σ ≤ τ and idG : (G, τ ) → (G, σ) semitopological imply τ = σ. Let G be the class of all topological groups. Remark 1.10. (a) Obviously K-minimality implies AK-completeness and K-minimality coincides with AK-completeness whenever all groups in K ⊆ G are abelian. (b) Moreover A-completeness coincides with AG-completeness. So Problem A can be seen as a particular case of Problem B, namely with K = G. (c) If K ⊆ K′ are classes of topological groups, then for every G ∈ K AK′ - complete implies AK-complete. In particular, if K ⊆ G and G ∈ K, then G A-complete implies G AK-complete. Clearly AH-completeness is a generalization of minimality, since H-minima- lity is precisely the usual minimality, which is intensively studied, as noted in Section 1.1. This is a strict generalization as shown by Example 6.1. A topological group G has small invariant neighborhoods (i.e., G is SIN ) if G is thin (i.e., it has a local base at eG of neighborhoods invariant under con- jugation). We prove that a topological group, which is SIN and AH-complete, is A-complete if and only it has trivial center (see Remark 6.8). In particular, if G is a group with trivial center, its discrete topology is AH-complete if and only if G ∈ T (see Corollary 6.6). So also in this case Tăımanov groups play a central role. Moreover we give an example of a small class K in which each element is AK-complete (see Example 6.14). This class is built on the Heisenberg group HR :=   1 R R 0 1 R 0 0 1   , that is the group of upper unitriangular 3 × 3 matrices over R, endowed with different group topologies. The group HR is nilpotent of class 2. In a forthcoming paper [6] we extend this example for generalized Heiseberg groups, that is, the group of upper unitriangular 3 × 3 matrices over a unitary ring A. In Example 7.5 we resolve negatively item (b) of Problem C. Moreover The- orem 7.2 answers partially (a), in the case when the topologies on the domain and on the codomain are the discrete and the indiscrete one respectively. Since we consider the trivial topologies, the condition that we find is exclusively al- gebraic. Indeed we prove that idG : (G, δG) → (G, ιG) is composition of n semitopological isomorphisms if and only if G is nilpotent of class ≤ n, where n ∈ N+. Arnautov’s problems on semitopological isomorphisms 91 Notation and terminology. We denote by R, Q, Z, P, N and N+ respectively the field of real numbers, the field of rational numbers, the ring of integers, the set of primes, the set of natural numbers and the set of positive integers. Let G be a group and x, y ∈ G. We denote by [x, y] the commutator of x and y in G, that is [x, y] = xyx−1y−1 and for x ∈ G and a subset Y of G let [x, Y ] = {[x, y] : y ∈ Y }. More in general, if H and K are subgroups of G, let [H, K] = 〈[h, k] : h ∈ H, k ∈ K〉, and in particular the derived G′ of G is G′ = [G, G], that is, the subgroup of G generated by all commutators of elements of G. The center of G is Z(G) = {x ∈ G : xg = gx, ∀g ∈ G} and for g ∈ G the centralizer of g in G is cG(g) = {x ∈ G : xg = gx}. The diagonal map ∆ : G → G×G is defined by ∆(g) = (g, g) for every g ∈ G. If H is another group, we denote by p1 : G × H → G and p2 : G × H → H the canonical projections on the first and the second component respectively. If f : G → H is a homomorphism, denote by Γf the graph of f , that is the subgroup Γf = {(g, f (g)) : g ∈ G} of G × H. If τ is a group topology on G then denote by V(G,τ )(eG) the filter of all neighborhoods of eG in (G, τ ) and by Bτ a base of V(G,τ )(eG). If X is a subset of G, X τ stands for the closure of X in (G, τ ). If N is a normal subgroup of G and π : G → G/N is the canonical projec- tion, then τq is the quotient topology of τ in G/N . Moreover Nτ denotes the subgroup {eG} τ . The discrete topology on G is δG and the indiscrete topology on G is ιG. For undefined terms see [12, 13]. 2. Properties of semitopological isomorphisms In the next remark we discuss the possibility to consider only the case of one group G endowed with two different topologies τ ≥ σ taking idG : (G, τ ) → (G, σ) as the continuous isomorphism: Remark 2.1. Let (G, τ ), (H, η) be topological groups and f : (G, τ ) → (H, η) a continuous isomorphism. Consider the topology σ = f −1(η) on G. This topology σ is coarser than τ and so idG : (G, τ ) → (G, σ) is a continuous isomorphism and (G, σ) is topologically isomorphic to (H, η). In particular idG : (G, τ ) → (G, σ) is semitopological if and only if f : (G, τ ) → (H, η) is semitopological. Moreover the next proposition shows that semitopological is a “local” prop- erty, like the stronger property open. The proof is a simple application of Theorem 1.2. Proposition 2.2. Let G be a group and τ, σ group topologies on G such that σ ≤ τ . Then idG : (G, τ ) → (G, σ) is semitopological if there exists a τ -open subgroup N of G such that idG ↾N : (N, τ ↾N ) → (N, σ ↾N ) is semitopological. 92 D. Dikranjan and A. Giordano Bruno The following theorems show the stability of the class of semitopological isomorphisms under taking subgroups, quotients and products. Theorem 2.3 ([2, Theorems 7 and 8]). Let G be a group, σ ≤ τ group topolo- gies on G and suppose that idG : (G, τ ) → (G, σ) is semitopological. (a) If A is a subgroup of G, then idA : (A, τ ↾A) → (A, σ ↾A) is semitopo- logical. (b) If A is a normal subgroup of G, then idG/A : (G/A, τq) → (G/A, σq ) is semitopological. Theorem 2.4 ([2, Theorem 9], [14, Theorem 6.15]). Let {Gi : i ∈ I} be a family of groups and {τi : i ∈ I}, {σi : i ∈ I} families of group topologies such that σi ≤ τi are group topologies on Gi for every i ∈ I. Then idGi : (Gi, τi) → (Gi, σi) is semitopological for every i ∈ I if and only if ∏ i∈I idGi :∏ i∈I (Gi, τi) → ∏ i∈I (Gi, σi) is semitopological. The next lemma shows a cancellability property of compositions of semi- topological isomorphisms. Lemma 2.5 ([14, Theorem 6.11]). Let σ ≤ τ be group topologies on a group G. If idG : (G, τ ) → (G, σ) is semitopological, then for a group topology ρ on G such that σ ≤ ρ ≤ τ , idG : (G, τ ) → (G, ρ) is semitopological. In a particular case, that is for initial topologies, the converse implication of Theorem 2.3(b) holds true: Lemma 2.6. Let G be a group and N a normal subgroup of G. Let σ ≤ τ be group topologies on G/N and σi ≤ τi the respective initial topologies on G. Then idG : (G, τi) → (G, σi) is semitopological if and only if idG/N : (G/N, τ ) → (G/N, σ) is semitopological. In the next theorem we consider the particular cases when one of the two topologies on G is trivial: Theorem 2.7 ([2, Corollary 5], [14, Corollary 5.11]). Let G be a group and τ a group topology on G. Then: (a) idG : (G, δG) → (G, τ ) is semitopological if and only if cG(g) is τ -open for every g ∈ G; (b) idG : (G, τ ) → (G, ιG) is semitopological if and only if G ′ ≤ Nτ . Since Z(G) ⊆ cG(g) for every g ∈ G, by (a) idG : (G, δG) → (G, τ ) semi- topological implies Z(G) τ -open. The condition G′ ≤ Nτ in (b) is equivalent to say that G ′ is indiscrete endowed with the topology inherited from (G, τ ). Moreover, as noted in [14], it implies that (G, τ ) is SIN. For SIN groups condition (a) of Theorem 1.2 is always verified, since SIN groups are thin, so only condition (b) remains: Arnautov’s problems on semitopological isomorphisms 93 Proposition 2.8. Let G be a group and σ ≤ τ group topologies on G. Suppose that (G, τ ) is SIN. Then idG : (G, τ ) → (G, σ) is semitopological if and only if for every U ∈ V(G,τ )(eG) and for every g ∈ G there exists Vg ∈ V(G,σ)(eG) such that [g, Vg] ⊆ U . The next lemma gives a simple necessary condition of algebraic nature for a continuous isomorphism to be semitopological. Lemma 2.9. Let G be a group and σ ≤ τ group topologies on G, such that idG : (G, τ ) → (G, σ) is semitopological. Then [G, Nσ] ≤ Nτ . Proof. By Theorem 1.2, for every U ∈ V(G,τ )(eG) and every g ∈ G, there exists Vg ∈ V(G,τ )(eG) such that [g, Vg] ⊆ U . Consequently [g, Nσ] ⊆ U for every g ∈ G, so [g, Nσ] ⊆ Nτ for every g ∈ G and hence [G, Nσ] ≤ Nτ . � Corollary 2.10. Let G be a group and τ a group topology on G. If τ is Hausdorff, then idG : (G, τ ) → (G, ιG) is semitopological if and only if G is abelian. Proof. If idG : (G, τ ) → (G, ιG) is semitopological, by Lemma 2.9 G ′ ≤ Nτ = {eG} and hence G is abelian. If G is abelian every continuous isomorphism is semitopological. � In particular idG : (G, δG) → (G, ιG) is semitopological if and only if the group G is abelian. Proposition 2.11. Let G be a group and σ ≤ τ group topologies on G, such that idG : (G, τ ) → (H, σ) is semitopological. If Z(G) = {eG} and τ is Haus- dorff, then σ is Hausdorff as well. Proof. Since Nτ = {eG} and [G, Nσ] ≤ Nτ by Lemma 2.9, using the hypothesis Z(G) = {eG} we conclude that Nσ = {eG}. � 3. Almost trivial topologies In this section we introduce a class of group topologies containing the trivial ones and with nice stability properties; moreover we extend Theorem 2.7 to this class. Definition 3.1. [14, Definition 5.13] A topological group (G, τ ) is almost trivial if Nτ is open in (G, τ ). Since in this case τ is completely determined by the normal subgroup N := Nτ of G, we denote an almost trivial topology on G by ζN , underling the role of the normal subgroup. Every group topology on a finite group is almost trivial and every almost trivial group is SIN. For example, for a group G, the discrete and the indiscrete topologies (i.e., the so-called trivial topologies) are almost trivial, with δG = ζ{eG} and ιG = ζG. This justifies the term used in Definition 3.1. 94 D. Dikranjan and A. Giordano Bruno Lemma 3.2. Let G be a simple non-abelian group and let τ be a group topology on G. Then either Nτ = G or Nτ = {eG}, that is, either τ = ιG or τ is Hausdorff, respectively. If τ is almost trivial, then τ is either discrete or indiscrete. The almost trivial topologies help also to express in simple terms topological properties: Remark 3.3. Given a topological group (G, τ ) and a normal subgroup N of G, it is possible to consider the group topology obtained “adding” to the open neighborhoods also N (since it is normal, it suffices to add N to the prebase of the neighborhoods and all the intersections U ∩ N , with U ∈ V(G,τ )(eG), give the neighborhoods of eG in the new topology). This new topology is sup{τ, ζN }. For example, if G is a group and τ its profinite topology, with Bτ = {Nα}α, where the Nα are all the normal subgroups of G of finite index, then τ = supα ζNα . More in general, if τ is a linear topology on G, that is Bτ = {Nα}α, where Nα are normal subgroups of G, then τ = supα ζNα . If (G, τ ) is a topological group, let τ̄ denote the quotient topology of (G, τ ) with respect to the normal subgroup Nτ , which is indiscrete. Then τ̄ is Haus- dorff. Moreover (G, τ ) is almost trivial if and only if (G/Nτ , τ̄ ) is discrete. Analogously it is possible to consider the case when a topological group (G, τ ) has a discrete normal subgroup D such that (G/D, τq) is indiscrete. For groups with this property we have a strong consequence: Lemma 3.4. Let (G, τ ) be a topological group such that D is a discrete normal subgroup of (G, τ ) and (G/D, τq) is indiscrete. Then (G, τ ) ∼= D × Nτ , where D is discrete and Nτ is indiscrete. In particular τ is almost trivial. Proof. Pick a symmetric neighborhood W of eG in G such that W 3∩D = {eG}. Since (G/D, τq) is indiscrete, D is dense in G, so G = DW . Let w1, w2 ∈ W . Then there exists d ∈ D such that w1w2 ∈ dW . Let w1w2 = dw for some w ∈ W . Then d = w1w2w −1 ∈ W 3 ∩ D = {eG}. So w1w2 = w ∈ W . Since W is symmetric, this proves that W is an open subgroup of M with W ∩ D = {eG}. Hence the restriction of the canonical projection G → G/D to W gives a topological isomorphism W ∼= (G/D, τq). This shows that W is an indiscrete group. Since Nτ ≤ W is closed, we deduce that W = Nτ . This proves that Nτ is open in τ and that (G, τ ) ∼= D × Nτ . � 3.1. Permanence properties of the almost trivial topologies. The as- signment N 7→ ζN defines an order reversing bijection between the complete lattice N (G) of all normal subgroups of a group G and the complete lattice AT (G) of all almost trivial group topologies on G. Let us note that the com- plete lattice AT (G) is not a sublattice of the complete lattice T (G) of all group topologies on G. Indeed, the meet of a family {ζNi : i ∈ I} in AT (G) is simply ζ⋂ i∈I Ni , whereas the meet of a family {ζNi : i ∈ I} in T (G) is the group topol- ogy having as prebase of the neighborhoods at eG the family {ζNi : i ∈ I} (in Arnautov’s problems on semitopological isomorphisms 95 other words, the latter topology may be strictly weaker than the former one in case I is infinite). The next lemma shows, among others, that the class of almost trivial groups is closed under taking subgroups and quotients. Lemma 3.5. Let (G, ζN ) be an almost trivial group, where N is a normal subgroup of G. (a) For every subgroup H of G: (a1) the topology induced on H by ζN is almost trivial and coincides with ζH∩N ; (a2) the following conditions are equivalent: (i) H is ζN -open; (ii) H is ζN -closed; (iii) H ≥ N . (b) For every normal subgroup N0 of G the quotient topology of ζN on G/N0 is almost trivial and coincides with ζN0N/N0 . Remark 3.6. In connection to item (a1) of the previous lemma notice that if H is an open subgroup of a topological group G and H is almost trivial, then also G is almost trivial. Now we show that the class of almost trivial groups is stable also with respect to taking finite products. Lemma 3.7. Let G1, G2 be groups and N1, N2 normal subgroups of G1, G2 respectively. Then ζN1 × ζN2 = ζN1×N2 on G1 × G2. The next lemma follows directly from the definitions. Lemma 3.8. Let G be a topological group and N an indiscrete normal subgroup of G such that G/N is almost trivial. Then G is almost trivial. We want to generalize this lemma and we need the following concept. Definition 3.9. For a class of topological groups P one says that P has the three space property, if a topological group G belongs to P whenever N ∈ P and G/N ∈ P for some normal subgroup N of G. For example the class of all discrete groups and the class of all indiscrete groups have the three space property. So the next result shows that the class of all almost trivial groups is the smaller class with the three space property containing all discrete and all indiscrete groups. Proposition 3.10. The class of almost trivial groups has the three space prop- erty. Proof. We have to prove that, in case G is a group and N a normal subgroup of G, if τ is a group topology on G such that (N, τ ↾N ) and (G/N, τq ) are almost trivial, then (G, τ ) is almost trivial. Let M be the normal subgroup of G containing N such that M/N = Nτq . Then M/N is indiscrete and open in G/N . Consequently, M is open in (G, τ ). To end the proof we need to verify that M is almost trivial (see Remark 3.6). 96 D. Dikranjan and A. Giordano Bruno If τ ↾N is Hausdorff, equivalently it is discrete, since it is almost trivial, and by Lemma 3.4 M is almost trivial. So we consider now the general case. The subgroup N1 := Nτ ∩ N is the closure of {eG} in M . Then N1 is a normal subgroup of N . Now the normal subgroup N/N1 of the Hausdorff quotient group M/N1 is almost trivial and consequently discrete. Moreover, the quotient (M/N1)/(N/N1) ∼= M/N is indiscrete. So by the previous case the group M/N1 is almost trivial. Since the group N1 is indiscrete, we can conclude with Lemma 3.8. � 3.2. Semitopological isomorphisms between almost trivial topologies. Since every almost trivial group is SIN, it is possible to apply Proposition 2.8 instead of Theorem 1.2 to verify if a continuous isomorphism is semitopological. In case the topology on the domain or that on the codomain is almost trivial, the conditions of Theorem 1.2 become simpler: Proposition 3.11. Let (G, σ) be a topological group and let σ ≤ τ be group topologies on G. (a) If τ is almost trivial, then idG : (G, τ ) → (G, σ) is semitopological if and only if for every g ∈ G there exists Vg ∈ V(G,σ)(eG) such that [g, Vg] ⊆ Nτ . (b) If σ is almost trivial, then idG : (G, τ ) → (G, σ) is semitopological if and only if Nσ is U -thin for every U ∈ V(G,τ )(eG) and [G, Nσ] ≤ Nτ . Proof. (a) follows from Proposition 2.8. (b) The necessity of the condition that Nσ is U -thin for every U ∈ V(G,τ )(eG) follows from Theorem 1.2, while the necessity of [G, Nσ] ≤ Nτ follows from Lemma 2.9. The sufficiency of the two conditions is a consequence of Theorem 1.2. � If τ is Hausdorff in this proposition, then (b) becomes N ≤ Z(G). So we have the following corollary, which can be also seen as a consequence of Proposition 2.11. Corollary 3.12. Let G be a group. If τ is a Hausdorff group topology on G, then for every non-central τ -open subgroup N of G idG : (G, τ ) → (G, ζN ) is not semitopological. Combining together the two items of Proposition 3.11 we have precisely the following corollary, which is the “almost trivial version” of Theorem 1.2. Furthermore it shows that the necessary condition of Lemma 2.9 becomes also sufficient in the case of almost trivial topologies. Corollary 3.13. [14, Lemma 5.15] Let G be a group and ζN ≥ ζL almost trivial group topologies on G. Then idG : (G, ζN ) → (G, ζL) is semitopological if and only if [G, L] ≤ N . The next example is a consequence of this corollary. Arnautov’s problems on semitopological isomorphisms 97 Example 3.14. Let G be a group and ζN an almost trivial group topology on G. Consider (G, δG) idG −−→ (G, ζN ) idG −−→ (G, ιG). Then: (a) idG : (G, δG) → (G, ζN ) is semitopological if and only if N ≤ Z(G); (b) idG : (G, ζN ) → (G, ιG) is semitopological if and only if G ′ ≤ N . On a group G it is possible to consider the almost trivial topology generated by G′, that is ζG′ . A group G is perfect if G = G ′, and G is perfect if and only if ζG′ = ιG. Remark 3.15. With this topology generated by the derived group, we can write again Theorem 2.7(b) as: Let G be a group and τ a group topology on G. Then idG : (G, τ ) → (G, ιG) is semitopological if and only if τ ≤ ζG′ . Remark 3.16. Let N be a normal subgroup of a group G and let ζN be the respective almost trivial topology on G. Let τ be a group topology on G. Then idG : (G, ζNτ ) → (G, τ ) is con- tinuous. Moreover, if ζL is another almost trivial topology on G such that idG : (G, ζL) → (G, τ ) is continuous, then idG : (G, ζL) → (G, ζNτ ) is continu- ous. (G, ζNτ ) // (G, τ ) (G, ζL) :: u u u u u u u u u eeK K K K K Consequently idG : (G, ζL) → (G, τ ) semitopological implies idG : (G, ζL) → (G, ζNτ ) semitopological by Lemma 2.5, that is, [G, Nτ ] ≤ L by Corollary 3.13. 4. Tăımanov groups Let F ∈ [G]<ω be a finite subset of G and cG(F ) = ⋂ x∈F cG(x) the centralizer of F in G. Then C = {cG(F ) : F ∈ [G] <ω} is a family of subgroups of G closed under finite intersections. Then the Tăımanov topology TG has C as local base at eG, that is BTG = C. We collect in the next lemma the first properties of this topology. Lemma 4.1. Let G be a group. Then: (a) NTG = Z(G); (b) TG is Hausdorff if and only if Z(G) = {eG}; (c) G is abelian if and only if TG = ιG; (d) in case G is finitely generated, TG is almost trivial; in particular TG = δG if and only if Z(G) is trivial. 98 D. Dikranjan and A. Giordano Bruno 4.1. Permanence properties of the class T. The following results show that the Tăımanov topology has nice properties. The next proposition proves that it is a functorial topology with respect to continuous surjective homomor- phisms. Proposition 4.2. Let G be a group. Then every surjective homomorphism f : (G, TG) → (H, TH ) is continuous. Proof. Let h ∈ H and consider g ∈ G such that f (g) = h. Then f (cG(g)) ⊆ cH (h). This proves the continuity of f : (G, TG) → (H, TH ). � On the other hand, the next example shows that the Tăımanov topology is not functorial with respect to open surjective homomorphisms. Example 4.3. For HZ :=   1 Z Z 0 1 Z 0 0 1   the group of upper unitriangular 3×3 matrices over Z, the canonical projection π : (HZ, THZ ) → (HZ/Z(HZ), THZ/Z(HZ)) is not open. Indeed, since HZ/Z(HZ) =: G is abelian, TG = ιG by Lemma 4.1(c). More- over note that G ∼= Z × Z. Let h =   1 0 0 0 1 1 0 0 1   ∈ HZ. Then cHZ (h) =   1 0 Z 0 1 Z 0 0 1   and π(cHZ (h)) ∼= {0} × Z, which is not open in (G, ιG). Lemma 4.4. Let G = ∏ i∈I Gi. Then ∏ i∈I TGi ≤ TG. If I = {1, . . . , n} is finite, then TG1 × . . . × TGn = TG. Proof. Since all the canonical projections πi : (G, TG) → (Gi, TGi ) are contin- uous by Proposition 4.2, ∏ i∈I TGi ≤ TG. Suppose now that I = {1, . . . , n} is finite. If F is a finite subset of G1 × . . . × Gn, then it is contained in a finite subset of the form F1 × . . . × Fn, where each Fi is a finite subset of Gi for i = 1, . . . , n. Moreover cG(F1 × . . . × Fn) = cG1 (F1) × . . . × cGn (Fn). This proves that TG = TG1 × . . . × TGn . � Proposition 4.5. (a) If G ∈ T, then Z(G) = {eG}. (b) If G ∈ Tt, then G is perfect. Proof. (a) Follows from Lemma 4.1(b). (b) Since G/G′ is abelian and in T, G/G′ is trivial in view of Lemma 4.1(c), that is G = G′. � It follows from (a) that every non-trivial abelian group G 6∈ T. The next result about products is a consequence of Lemma 4.4. Proposition 4.6. The class T is closed under taking finite products. Arnautov’s problems on semitopological isomorphisms 99 Proof. Let G1, G2 ∈ T and G := G1 × G2. By Lemma 4.4 TG = TG1 × TG2 and so TG = δG, that is G ∈ T. This can be extended to all finite products. � The next example in particular shows that T is not closed under taking quotients and subgroups since the groups in (b) and (c) have abelian quotients (so they are not in Tt) and non-trivial abelian subgroups. Example 4.7. (a) A finite group G ∈ T if and only if Z(G) = {eG}. This follows from Lemma 4.1, but can be also simply directly proved. (b) Let G = ( R∗ R 0 1 ) . Then G ∈ T. Indeed, for F = {( 2 0 0 1 ) , ( 1 1 0 1 )} cG(F ) = {eG}. (c) Every non-abelian free group F (X) of rank > 1 is in T. Indeed, for F = {a, b}, where a, b ∈ X are generators of F (X), cF (X)(F ) = {eF (X)}. This example shows also that the condition “surjective” in Proposition 4.2 cannot be removed: if G is one of the groups in (b) or (c), then G has some non-trivial abelian subgroup H. Since H is abelian, TH = ιH , while TG = δG. Consequently the injective homomorphism (H, TH ) → (G, TG) is far from being continuous. Remark 4.8. Since non-abelian free groups of rank > 1 are Tăımanov (as shown in Example 4.7(c)), • there exist arbitrarily large Tăımanov groups; moreover, every non- abelian subgroup of a non-abelian free group is Tăımanov, being free [23]; • every group is quotient of a Tăımanov group, since every group is quo- tient of a non-abelian free group of rank > 1 [23]. It is not clear if this holds also for subgroups: Question 4.9. Is every group subgroup of a Tăımanov group? Theorem 4.11 answers positively the question in the abelian case. For an abelian group G and p ∈ P in what follows we denote by rp(G) the p-rank of G. Lemma 4.10. Let G be an abelian group with r2(G) = 0. Then there exists H ∈ T such that G ≤ H and [H : G] = 2. Proof. Let f : G → G be defined by f (x) = −x for every x ∈ G. Moreover let H := G ⋊ 〈f〉 (⋊ denotes the semidirect product). Then cH (0, f ) = 〈(0, f )〉 and (0, f ) 6∈ cH (g, idG) for every g ∈ G \ {0}. Consequently for F = {(g, idG), (0, f )}, with g ∈ G \ {0}, cH (F ) = {eH}, that is H ∈ T. Since f has order 2, G has index 2 in G. � Theorem 4.11. For every abelian group G there exists a group H ∈ T con- taining G as a subgroup and such that |H| = ω · |G|. 100 D. Dikranjan and A. Giordano Bruno Proof. Let G be an abelian group. Then G ⊆ D(G) = G1⊕G2, where r2(G1) = 0 and r2(G2) = r2(D(G)). Then there exists H1 ∈ T such that G1 ≤ H1 and |H1| = 2 · |G1| by Lemma 4.10. Now consider G2 = ⊕ r2(G) Z(2∞). If r2(G) ≤ ω, then G2 is contained in ⊕ ω Z(2 ∞). Let then σ be the shift ⊕ ω Z(2 ∞) → ⊕ ω Z(2 ∞) defined by (xn)n 7→ (xn−1)n for every (xn)n ∈ ⊕ ω Z(2 ∞). Then σn has no non-zero fixed point for every n ∈ Z, n 6= 0. Claim. Let G be an abelian group and let f be an automorphism of G such that f n has no non-zero fixed point for every n ∈ Z, n 6= 0. Then there exists H ∈ T such that G ≤ H and |H| = ω · |G|. Proof of the claim. Let H := G ⋊ 〈f〉. Then cH (0, f ) = 〈(0, f )〉 and (0, f ) 6∈ cH (g, idG) for every g ∈ G \ {0}. Con- sequently for F = {(g, idG), (0, f )}, with g ∈ G \ {0}, cH (F ) = {eH}, that is H ∈ T. Since f has infinite order |H| = ω · |G|. � By the claim there exists a group H2 ∈ T such that G2 ≤ ⊕ ω Z(2 ∞) ≤ H2 and |H2| = ω. Suppose that r2(G) ≥ ω. Then G2 ∼= ⊕ r2(G) ( ⊕ ω Z(2 ∞)). Let σ̃ : ⊕ r2(G) ( ⊕ ω Z(2 ∞)) → ⊕ r2(G) ( ⊕ ω Z(2 ∞)) be defined by σ̃ ↾⊕ ω Z(2∞)= σ. Then σ̃n has no non-zero fixed point for every n ∈ Z, n 6= 0, and again the claim gives a group H2 ∈ T that contains G2 as a subgroup and such that |H2| = |G2|. Let H := H1 ⊕ H2. By Proposition 4.6 H ∈ T. Moreover |H| = ω · |H1| · |H2| = ω · |G1| · |G2| = ω · |G|. � Lemma 4.13 shows that to prove that a group is Tăımanov it suffices to consider a convenient quotient with a finite normal subgroup and check whether it is Tăımanov. Claim 4.12. Let G be a group with Z(G) = {eG}. If there exists a finite subset F of G such that cG(F ) is finite, then there exists another finite subset F1 ⊇ F of G such that cG(F1) = {eG}. In particular G ∈ T. Proof. Let cG(F ) = {eG, g1, . . . , gn}. Since Z(G) = {eG}, for every i ∈ {1, . . . , n} there exists hi ∈ G such that [gi, hi] 6= eG; in particular gi 6∈ cG(hi). Let F1 = F ∪ {h1, . . . , hn}. Then gi 6∈ cG(F1) for every i ∈ {1, . . . , n}. Since cG(F1) ⊆ cG(F ) = {eG, g1, . . . , gn}, this proves that cG(F1) = {eG}. � Lemma 4.13. Let G be a group with Z(G) = {eG} and let N be a normal finite subgroup of G such that G/N ∈ T. Then G ∈ T. Proof. Let F1 be a finite subset of G/N such that cG/N (F1) = {eG/N }. Let π : G → G/N be the canonical projection and let F be a finite subset of G such that π(F ) = F1. Since π(cG(F )) ⊆ cG/N (F1) = {eG/N }, cG(F ) ⊆ N . Since N is finite, Claim 4.12 applies to conclude that G ∈ T. � Arnautov’s problems on semitopological isomorphisms 101 The next is an example of a totally Tăımanov group. Example 4.14. We denote by G := SO3(R) the group of all orthogonal ma- trices 3 × 3 with determinant 1 and coefficients in R. Then G ∈ T. Since G is simple, G ∈ Tt. Indeed, G = ⋃ α Tα, where Tα ∼= T and Tα is generated by an element α of G, that is, 〈α〉 = Tα. Moreover cG(α) contains Tα as a finite index subgroup and for α, β ∈ G with α 6= β and Tα 6= Tβ, Tα ∩Tβ is finite. Then cG(α)∩cG(β) is finite. By Claim 4.12 G ∈ T. 4.2. The permutations groups. For a set X, x ∈ X and a subgroup H of S(X) let OH (x) := {h(x) : h ∈ H}, Stab x := {ρ ∈ S(X) : ρ(x) = x}, and Sx := Stab x ∩ H. Moreover H induces a partition of X, that is X = ⋃ x∈RH OH (x), where RH ⊆ X is a set of representing elements. If τ ∈ S(X), then Stab x = (Stab τ (x))τ . Remark 4.15. Let X be a set and H a subgroup of S(X). If τ ∈ NS(X)(H), then: (a) τ (OH (x)) = OH (τ (x)); (b) Sx = (Sτ (x)) τ ; indeed, Sx = Stab x ∩ H = (Stab τ (x)) τ ∩ Hτ = (Stab τ (x) ∩ H)τ = (Sτ (x)) τ ; (c) τ induces a permutation τ̃ of RH . Indeed, τ (OH (x)) = OH (τ (x)) by (a); so we can define τ̃ (x) = y, where y ∈ RH is the representing element of OH (τ (x)). Then cS(X)(H) =    τ ∈ ⋂ x∈RH \supp τ̃ NH (Sx) · Stab x : [H, τ ] ⊆ ⋂ x∈supp τ̃ Sx    . We describe the subgroups of S(X) with trivial centralizer: Lemma 4.16. Let X be a set and H a subgroup of S(X). Then cS(X)(H) = {idX} if and only if the following conditions hold: (a) Sx = NH (Sx) for every x ∈ RH , and (b) Sx and Sy are not conjugated in H for every x, y ∈ RH with x 6= y. Proof. Let τ ∈ cS(X)(H)\{idX}. There exists x ∈ RH such that y := τ (x) 6= x. Indeed, if τ (x) = x for every x ∈ RH , then for every z ∈ X, there exist h ∈ H and x ∈ RH such that z = h(x), and so τ (z) = τ (h(x)) = h(τ (x)) = h(x) = z. By Remark 4.15(a,b) τ (OH (x)) = OH (τ (x)) and Sx = (Sy) τ = Sy. 102 D. Dikranjan and A. Giordano Bruno If y ∈ OH (x), then τ ↾OH (x): OH (x) → OH (x) is a bijection and y = h0(x) for some h0 ∈ H; then τ (h(x)) = h(τ (x)) = hh0(x) for every h ∈ H. Let h ∈ Sx. Since τ is well-defined, h(x) = x implies hh0(x) = h0(x), that is (h0) −1hh0(x) = x. This is equivalent to h h0 ∈ Sx, that is h0 ∈ NH (Sx). But h0 6∈ Sx and this contradicts (a). Suppose now that y 6∈ OH (x) and so OH (x) ∩ OH (y) = ∅. Let z ∈ RH ∩ OH (y). Then y = h0(z) for some h0 ∈ H. By Remark 4.15(b) Sz = (Sy) h0 = (Sx) h0 and this contradicts (b). Assume that there exists h0 ∈ NH (Sx)\Sx for some x ∈ RH . Let τ : X → X be defined by τ (x) = h0(x), τ (h(x)) = hh0(x) for every h ∈ H and τ (y) = y for every y ∈ X \ OH (x). This τ is well-defined. Indeed, if h1(x) = h2(x) for some h1, h2 ∈ H, that is, h −1 2 h1 ∈ Sx; then h1h0(x) = h2h0(x), equivalently h−10 (h −1 2 h1)h0(x) = x, that is, h −1 0 (h −1 2 h1)h0 ∈ Sx, which holds true by the hypothesis that h0 ∈ NH (Sx). Moreover, it is possible to check that τ ∈ S(X). By the definition τ h = hτ for every h ∈ H and so idX 6= τ ∈ cS(X)(H). Suppose that Sx = (Sz ) h0 for some x, z ∈ RH and h0 ∈ H. Then for y = h−10 (z) ∈ OH (z) we have Sy = (Sz) h0 = Sx by Remark 4.15(b). Define τ : X → X as τ (x) = y, τ (h(x)) = h(y) for every h ∈ H and τ (w) = w for every w ∈ X \ OH (x). Then τ is well-defined; indeed, if h1(x) = h2(x) for some h1, h2 ∈ H, that is, h −1 2 h1 ∈ Sx, then h1(y) = h2(y), equivalently, h −1 2 h1 ∈ Sy, which holds true since Sx = Sy. Moreover, it is possible to check that τ ∈ S(X). By the definition τ h = hτ for every h ∈ H and so idX 6= τ ∈ cS(X)(H). � Proposition 4.17. For a cardinal κ the following conditions are equivalent: (a) there exists a set X with |X| = κ and S(X) ∈ T; (b) there exists a set X with |X| = κ such that there exists a finitely gen- erated subgroup H of S(X) such that Sx = NH (Sx) for every x ∈ RH and Sx, Sy are not conjugated for every x, y ∈ RH with x 6= y. If κ > ω, then the following condition is equivalent to the previous: (c) there exists a finitely generated group H admitting a family S = {Sα : α < κ} of subgroups of H such that Sα = NH (Sα) for every α < κ. Proof. (a)⇔(b) The condition S(X) ∈ T is equivalent to the existence of a finite subset F of S(X) such that cS(X)(F ) = {idX}. Let H = 〈F 〉. Then cS(X)(H) = cS(X)(F ) and so equivalently cS(X)(H) = {idX}. By Lemma 4.16 we have the conclusion. (b)⇒(c) Since κ > ω, and each OH (x) is countable, |RH| = κ. So {Sx : x ∈ RH} is the family requested in (c). (c)⇒(b) Since κ > ω and H is countable, we can suppose that S has the property that Sα and Sβ are not conjugated in H for every α, β < κ with α 6= β. Indeed, every subgroup Sα of H has at most countably many conjugated subgroups in H, so we can restrict the family S taking only one element for every class of conjugation, finding a subfamily of the same cardinality κ as S. Arnautov’s problems on semitopological isomorphisms 103 Define Xα := {hSα : h ∈ H} for every α < κ and X := ⋃ α<κ Xα. Moreover let xα := idH Sα ∈ Xα for every α > κ. In particular |X| = κ. Moreover H acts on X by multiplication on the left and OH (xα) = Xα for every α < κ. There exists a group homomorphism ϕ : H → S(X); let H̃ := ϕ(H) ≤ S(X). Then H̃ is finitely generated and the action of H̃ on X is the same as the action of H on X. Then O H̃ (xα) = Xα for every α < κ and RH̃ = {xα : α < κ}. Moreover ϕ(Sα) = Stab xα ∩ H̃ =: Sxα . Since Sα = NH (Sα) for every α < κ and Sα and Sβ are not conjugated for every α < β < κ, it is possible to prove that Sxα = NH (Sxα ) for every xα ∈ RH̃ and Sxα and Sxβ are not conjugated for every xα, xβ ∈ RH̃ with xα 6= xβ . So the properties in (b) are satisfied. � Theorem 4.18. Let X be a set with |X| > 2. (a) If |X| ≤ ω, then S(X) ∈ T. (b) If |X| > c, then S(X) 6∈ T. Proof. (a) Assume that 2 < |X| < ω. Since Z(S(X)) is trivial, S(X) ∈ T by Example 4.7(a). Assume that |X| = ω. We can suppose X = Z. Let H = 〈σ, τ〉, where τ = (−1, 1) and σ is the shift, that is σ(n) = n + 1 for every n ∈ Z. Then OH (0) = Z and so RH = {0}. Moreover S0 = 〈τ〉 and hence NH (S0) = S0. By Proposition 4.17 S(X) ∈ T. (b) Let H be a finitely generated subgroup of S(X). Since OH (x) is count- able for every x ∈ RH , |RH| = |X| > c. Since H is countable, it has at most c subgroups and so there exists a subset S of RH such that |S| > c and Sx = Sy for every x, y ∈ S. By Proposition 4.17 S(X) 6∈ T. � Question 4.19. Let X be a set. (a) Is S(X) ∈ T if |X| = ω1? (b) Is S(X) ∈ T if |X| = c? (c) Is S(X) ∈ T if ω < |X| ≤ c? Remark 4.20. Question 4.19 can be formulated in equivalent terms thanks to Proposition 4.17. Indeed, if X is a set of cardinality κ with ω < κ ≤ c, then S(X) ∈ T if and only if there exists a finitely generated group H admitting a family S = {Sα : α < κ} of subgroups of H such that Sα = NH (Sα) for every α < κ. So Question 4.19 becomes: does there exist a finitely generated group H with a “large” (i.e., of cardinality κ with ω < κ ≤ c) family of self-normalizing subgroups? 5. Problem A We start considering a stability property of the the class A of Arnautov groups. Theorem 5.1. The class A is closed under taking quotients. 104 D. Dikranjan and A. Giordano Bruno Proof. Let G ∈ A and let N be a normal subgroup of G. Let σ ≤ τ be group topologies on G/N such that idG/N : (G/N, τ ) → (G/N, σ) is semitopological. Then idG : (G, τi) → (G, σi) is semitopological by Lemma 2.6. Since G ∈ A, τi = σi and hence τ = σ. � In Section 5.2 we will comment the stability of A under taking subgroups and products. Example 5.2. (a) Obviously every indiscrete group G is A-complete. (b) Let G be a group. Let Gab = G/G ′ be the abelianization of G and endow Gab with the discrete topology and with the indiscrete topology: (G, ζG′ ) −−−−→ (Gab, δGab ) idG y yidGab (G, ιG) −−−−→ (Gab, ιGab ) If G 6= G′ then idGab is a semitopological non-open isomorphism, be- cause Gab is abelian, and idG is a semitopological non-open isomor- phism too, in view of Remark 3.15. So (G, ζG′ ) is not A-complete. (c) An abelian topological group G is A-complete if and only if G is indis- crete. In particular the only abelian Arnautov group is G = {eG} (as (G, δG) must be indiscrete). The next proposition generalizes the example in (b). Proposition 5.3. A topological group G with indiscrete derived group G′ is A-complete precisely when G is indiscrete. Proof. The conclusion follows from Remark 3.15. � Example 5.4. Let G be a group and τ a group topology on G. (a) If (G, τ ) is SIN, then it is A-complete if and only if for every group topology σ < τ on G there exist U ∈ V(G,τ )(eG) and g ∈ G such that [g, Vg] 6⊆ U for every Vg ∈ V(G,σ)(eG) (this follows from Proposition 2.8). (b) If (G, τ ) is Hausdorff and τ ≤ ζG′ (as already noted after Theorem 2.7, this condition yields τ SIN), then G is abelian and consequently τ > ιG implies that (G, τ ) is not A-complete (supposing that G is not a singleton). Proposition 5.5. Let G be a group and N a normal subgroup of G. Let τ be a group topology on G/N and τi the initial topology of τ on G. Then τ is A-complete if and only if τi is A-complete. Proof. Let idG/N : (G/N, τ ) → (G/N, σ) be semitopological, where σ ≤ τ is another group topology on G. By Lemma 2.6 also idG : (G, τi) → (G, σi) is semitopological and the hypothesis implies that τi = σi. Consequently τ = σ. Arnautov’s problems on semitopological isomorphisms 105 Suppose that τ is A-complete. Let σ < τi be another group topology on G and consider the quotient topology σq of σ on G/N . So we have the following situation: (G, τi) idG −−−−→ (G, σ) π y yπ (G/N, τ ) idG/N −−−−→ (G/N, σq ). Since σ < τi, it follows that Nσ ≥ Nτi = N . Consequently σ is the initial topology of σq and so σq < τ , otherwise σ = τi. By hypothesis idG/N : (G/N, τ ) → (G/N, σq ) is not semitopological. To conclude that also idG : (G, τi) → (G, σ) is not semitopological apply Theorem 2.3. � Corollary 5.6. Let G be a group and τ a group topology on G. Consider the quotient G/Nτ and the quotient topology τq of τ on G/Nτ . Then τ is A-complete if and only if τq is A-complete. Proof. Since τ is the initial topology of τq, it suffices to apply Proposition 5.5. � Now we give a necessary condition for a group to be Arnautov. Proposition 5.7. For a group G the following conditions are equivalent: (a) idG : (G, τ ) → (G, ιG) is semitopological for no group topology τ > ιG on G; (b) G is perfect. Proof. (a)⇒(b) Since idG : (G, ζG′ ) → (G, ιG) is a semitopological isomorphism by Theorem 2.7(b), our hypothesis (a) implies ζG′ = ιG and hence G = G ′. (b)⇒(a) Suppose G = G′; then ζG′ = ιG. If idG : (G, τ ) → (G, ιG) is a semitopological isomorphism, then τ ≤ ζG′ = ιG by Theorem 2.7(b), so τ = ιG. This means that idG is open. � Therefore, if a group G is Arnautov, then for every non-indiscrete group topology τ on G idG : (G, τ ) → (G, ιG) is not semitopological. In particular Proposition 5.7 implies that every Arnautov group is perfect. Corollary 5.8. Let G be a simple non-abelian group and τ a group topology on G. If τ > ιG, then idG : (G, τ ) → (G, ιG) is not semitopological. A consequence of these results is that every minimal simple non-abelian group (G, τ ) is A-complete. Indeed, if σ ≤ τ is another group topology on G and idG : (G, τ ) → (G, σ) is semitopological, then by Lemma 3.2 either σ is Hausdorff or σ = ιG. Since G is simple and non-abelian, G is perfect. Then Proposition 5.7 implies that σ is not indiscrete and so σ has to be Hausdorff. The minimality of τ yields that σ = τ . This consequence is improved by the next result. Proposition 5.9. If (G, τ ) is a minimal group and Z(G) = {eG}, then (G, τ ) is A-complete. 106 D. Dikranjan and A. Giordano Bruno Proof. Let σ ≤ τ be a group topology on G and suppose that idG : (G, τ ) → (G, σ) is semitopological. By Proposition 2.11 σ is Hausdorff and so σ = τ by the minimality of τ . � Example 5.10. Every simple finite non-abelian group G is an Arnautov group. Indeed, the only group topologies on G are the trivial ones and idG : (G, δG) → (G, ιG) is not semitopological by Corollary 5.8. The following remark could be used as a test to verify if a group is Arnautov. Remark 5.11. If G ∈ A, then for every group topology τ on G and for every normal subgroup N of G, • idG : (G, sup{τ, ζN }) → (G, τ ) is not semitopological if sup{τ, ζN } > τ ; • idG : (G, sup{τ, ζN }) → (G, ζN ) is not semitopological if sup{τ, ζN } > ζN . 5.1. When the discrete topology is A-complete. Remark 5.12. [14, Corollary 5.3] We can formulate Theorem 2.7(a) in terms of the Tăımanov topology: Let G be a group and σ a group topology on G. Then idG : (G, δG) → (G, σ) is semitopological if and only if σ ≥ TG, that is, Nσ ≤ NTG = Z(G). Consequently the Tăımanov topology is the coarsest topology σ on a group G such that idG : (G, δG) → (G, σ) is semitopological. So, since in this section we consider the case when the discrete topology is A-complete, we have to impose that the Tăımanov topology is discrete, that is, the group is Tăımanov. This also motivates Definition 1.8. The next theorem solves a particular case of Problem A, that is, it charac- terizes the groups for which the discrete topology is A-complete. Theorem 5.13. Let G be a group. Then δG is A-complete if and only if G ∈ T. Proof. Suppose that δG > TG. Then idG : (G, δG) → (G, TG) is semitopological by Remark 5.12. This proves that δG is not A-complete. Suppose that δG = TG. Let τ < δG be a group topology on G. Then idG : (G, δG) → (G, τ ) is not semitopological by Remark 5.12. This proves that δG is A-complete. � By Proposition 4.5(a) the equivalent conditions of this theorem imply that the group has trivial center. The next example shows that they can be strictly stronger than having trivial center. Moreover this is an example of a Tăımanov group which has an infinite non-abelian subgroup that is not Tăımanov. Example 5.14. Consider S(N) and let G := Sω(N) be the subgroup of S(N) of the permutations with finite support, that is Sω = ⋃∞ n=1 Sn. Then Z(G) = {eG}. If F is a finite subset of G, then there exists n ∈ N+ such that F ⊆ Sn and c(Sn) = S(N \ {1, . . . , n}) is infinite. Therefore TG < δG and so G 6∈ T. Arnautov’s problems on semitopological isomorphisms 107 Anyway in the finite case the three conditions are equivalent, as stated by Example 4.7(a). The next theorem characterizes the almost trivial topologies that are A- complete. It covers Theorem 5.13. Theorem 5.15. Let G be a group and N ⊳ G. Then (G, ζN ) is A-complete if and only if G/N ∈ T. Proof. Suppose that ζN is A-complete. Since ζN is the initial topology of δG/N , it follows that δG/N is A-complete by Proposition 5.5. By Theorem 5.13 this is equivalent to G/N ∈ T. Suppose now that G/N ∈ T. By Theorem 5.13 this is equivalent to say that δG/N is A-complete and so ζN is A-complete by Corollary 5.6. � The next theorem offers a relevant necessary condition for a group to be Arnautov: Theorem 5.16. If G ∈ A, then G ∈ Tt. Proof. The conclusion follows from Theorems 5.1 and 5.13. � So the next question naturally arises. Question 5.17. Does G ∈ Tt imply G ∈ A? We shall give a positive answer to this question in a particular case in Propo- sition 5.25. The next examples show that a group can admit two A-complete topologies that are one strictly finer than the other. Example 5.18. Let G := S(Z) and S := Sω(Z) > A := Aω (Z), which are the only proper normal subgroups of G. (a) The point-wise convergence topology T on G is A-complete: T is min- imal and Z(G) is trivial, so Proposition 5.9 applies. (b) The discrete topology δG is A-complete by Theorems 4.18 and 5.13. (c) We show that Z(G/A) = S/A and |S/A| = 2. The group S/A has only one non-trivial element, that is, S/A = 〈π(τ )〉, where π : G → G/A is the canonical projection and τ = (12) ∈ G. Indeed, if σ ∈ S and σ 6∈ A, then τ σ ∈ A and so π(σ) ∈ 〈π(τ )〉. Moreover τ 6∈ A. Since S/A is a non-trivial normal subgroup of G/A and it has size 2, it is central; since S/A is the unique non-trivial normal subgroup of G/A, S/A = Z(G/A). (d) It follows from (c) that G 6∈ T by Proposition 4.5(a). (e) By (d) ζA is not A-complete in view of Theorem 5.15, hence G 6∈ A. (f) Moreover it is possible to prove that G/S ∈ T. Consequently G/S ∈ Tt, being simple. This is an example of a group G which is not Arnautov but with δG A- complete. Moreover, since the subgroup of G generated by the shift σ is abelian 108 D. Dikranjan and A. Giordano Bruno and so not A-complete, while δG is A-complete, this example shows also that a subgroup of an A-complete group need not be A-complete. Example 5.19. Consider the group G := SO3(R). As shown by Example 4.14, G ∈ T. Consequently δG is A-complete by Theorem 5.13. Moreover the usual compact topology τ of G is A-complete, because τ compact implies minimal, Z(G) is trivial and so Proposition 5.9 applies. A first step to find an answer to Question 5.17 is to consider the following. Question 5.20. (a) Does S(Z)/Sω(Z) ∈ A? (b) Does SO3(R) ∈ A? 5.2. Totally Markov groups. Our aim is to provide examples of groups in A. The next results shows that for totally Markov groups the topologies are all almost trivial and so to verify if a continuous isomorphism of a totally Markov group is semitopological is simple, thanks to Corollary 3.13. Proposition 5.21. A group G ∈ Mt if and only if every group topology on G is almost trivial. Proof. Suppose that G ∈ Mt and let τ be a group topology on G. Then the quotient topology of τ on G/Nτ is Hausdorff and hence discrete, being G ∈ Mt. So Nτ is open in (G, τ ) and therefore τ is almost trivial. Suppose that the group G 6∈ Mt. Then there exists a normal subgroup N of G such that there exists a Hausdorff non-discrete group topology σ on G/N . Let π : G → G/N be the canonical projection and τ = π−1(σ). Therefore Nτ = N (because N = ⋂ {V : V ∈ V(G/N,σ)(eG/N )} in G/N ). Since σ is non-discrete N is not open and so τ is not almost trivial. � Proposition 3.10, together with Proposition 5.21, immediately implies that Mt is closed under extensions: Definition 5.22. For a class of abstract groups P one says that P is closed under extensions, if a group G belongs to P whenever N ∈ P and G/N ∈ P for some normal subgroup N of G. Moreover we have the same result for M: Theorem 5.23. The classes M and Mt are closed under extensions. In par- ticular, M and Mt are closed under finite direct products. Proof. That Mt is closed under extensions is a direct consequence of Proposi- tions 3.10 and 5.21. Suppose that the group G has a normal subgroup N such that N ∈ M and G/N ∈ M. We show that G ∈ M. To this end let τ be a Hausdorff group topology on G. Then τ ↾N = δN . Consequently: (i) N is closed in (G, τ ), and (ii) π : (G, τ ) → (G/N, τq ) is a local homeomorphism. Arnautov’s problems on semitopological isomorphisms 109 By (i) (G/N, τq ) is Hausdorff and so discrete. In view of (ii) τ = δG. � In view of Theorem 5.13, a necessary condition for A-completeness of δG for a group G is Z(G) = {eG}. For Markov groups also the converse implication holds: Corollary 5.24. Let G ∈ M. Then G ∈ T if and only if Z(G) = {eG}. Proof. If G ∈ T, apply Theorem 5.13. Suppose Z(G) = {eG}. Then TG is Hausdorff by Lemma 4.1(b) and so TG = δG. � In the following proposition we characterize totally Markov groups which are A-complete or Arnautov. In particular it shows that for a totally Markov group it is equivalent to be Arnautov and to be totally Tăımanov, which is precisely the answer to Question 5.17 in the particular case of totally Markov groups. Proposition 5.25. Let G ∈ Mt. (a) If τ is a group topology on G, the following conditions are equivalent: (i) (G, τ ) is A-complete; (ii) G/Nτ ∈ T; (iii) for every N ⊳ G, if [G, N ] ≤ Nτ ≤ N , then N = Nτ . (b) The following conditions are equivalent: (i) G ∈ A; (ii) G ∈ Tt; (iii) Z(G/N ) = {eG/N } for every N ⊳ G; (iv) [G, N ] = N for every N ⊳ G. Proof. (a) The equivalence (i)⇔(ii) follows from Lemma 5.21 and Theorem 5.15. The equivalence (i)⇔(iii) follows from Lemma 5.21 and Corollary 3.13. (b) The equivalence (i)⇔(ii) follows from (a) and the equivalence (ii)⇔(iii) follows from Corollary 5.24. (iii)⇒(iv) Let N be a normal subgroup of G. Then [G, N ] is a normal subgroup of G and Z(G/[G, N ]) is trivial by hypothesis. Since N/[G, N ] ≤ Z(G/[G, N ]) also N/[G, N ] is trivial, that is N = [G, N ]. (iv)⇒(i) By Lemma 5.21 every group topology on G is almost trivial. So let L be a normal subgroup of G. For every normal subgroup N of G such that [G, N ] ≤ L ≤ N , N = L because [G, N ] = N by hypothesis. This proves that ζL is A-complete by (a). Consequently G ∈ A. � This proposition covers Example 5.10. Corollary 5.26. (a) A finite group G ∈ A if and only if G ∈ Tt. (b) For every G ∈ M simple, G ∈ A. In Example 5.18 we have seen that S(Z) 6∈ A, but S(Z)/Sω(Z) ∈ Tt. In relation to Question 5.20 we consider the following, which has also its own interest. In Example 4.14 we have seen that SO3(R) ∈ Tt, but clearly SO3(R) 6∈ M. 110 D. Dikranjan and A. Giordano Bruno Question 5.27. Does S(Z)/Sω(Z) ∈ M? A positive answer to this question would imply that S(Z)/Sω(Z) ∈ A, that is a positive answer to Question 5.20, in view of Corollary 5.26(b), since S(Z)/Sω(Z) is simple. From another point of view, in order to answer Ques- tion 5.27, it is possible to consider first Question 5.20 which involves a weaker condition. Example 5.28. Let V = (Fpm ) n, where m, n ∈ N+, p ∈ P and (n, p m −1) = 1. Define G to be the semidirect product of SL(V ) and V . Then [G, V ] = V . Moreover every normal subgroup of G contains V and so, since SL(V ) is simple, V is the unique non trivial normal subgroup of G. Then G ∈ A by Corollary 5.26(a). Example 5.29. (a) Corollary 5.26(b) provides an example of an infinite Arnautov group. Indeed Shelah [24] constructed a simple Markov (hence totally Markov) group M under CH. (b) The group M contains a subgroup isomorphic to Z, which is abelian and so not in A. (c) In general a totally Markov group need not be an Arnautov group, that is, Mt 6⊆ A; for example G := M × Z(2) ∈ Mt but G 6∈ A. Item (b) of this example shows that A is not stable under taking subgroups. Question 5.30. Is A stable under taking (finite) direct products? And under taking (finite) powers? In the next example we give examples of Arnautov groups which are not simple. Moreover we see a particular case (that of Markov simple groups) in which finite powers of Arnautov groups are Arnautov. Example 5.31. Let M ∈ M be simple; by Corollary 5.26(b) M ∈ A. We show that M n ∈ Mt and also M n ∈ A, for every n ∈ N+. Since M ∈ M is simple, M ∈ Mt. By Theorem 5.23 M n ∈ Mt for every n ∈ N+. So M n ∈ A by Proposition 5.25(b): for every normal subgroup N of M n, N = M k for some k ≤ n up to topological isomorphisms, and consequently [M n, N ] = [M n, M k] = M k = N . The next are corollaries of Propositions 3.10 and 5.21. Corollary 5.32. Let G be a group and N1 ≤ N2 be normal subgroups of G with N2/N1 ∈ Mt. Then every group topology τ on G with ζN2 ≤ τ ≤ ζN1 is almost trivial. In particular, (a) if N2 ∈ Mt, then every group topology τ on G with τ ≥ ζN2 is almost trivial; and (b) if G/N1 ∈ Mt, then every group topology τ on G with τ ≤ ζN1 is almost trivial. Proof. (a) Since N2 ∈ Mt, by Proposition 5.21 τ ↾N2 is almost trivial. Moreover τq ≥ (ζN2 )q = δG/N2 on G/N2, and so τq = δG/N2 and in particular it is almost trivial. By Proposition 3.10 τ is almost trivial. Arnautov’s problems on semitopological isomorphisms 111 Obviously, N1 ≤ Nτ ≤ N2. Therefore, the quotient topology τq of (G, τ ) with respect to N1 satisfies δG/N1 ≥ τq ≥ ζN2/N1 . To the normal subgroup N2/N1 ∈ Mt of the group G/N1 and τq ≥ ζN2/N1 we apply (a) to claim that τq is almost trivial. Since τq was obtained from τ via a quotient with respect to the τ -indiscrete normal subgroup N1, by Lemma 3.8 τ is almost trivial. (b) Follows from the previous part. � Corollary 5.33. Let G be a group and N1 ≤ N2 be normal subgroups of G with [N2 : N1] finite. Then G admits only finitely many group topologies τ with ζN2 ≤ τ ≤ ζN1 and they are all almost trivial. Proof. Apply Corollary 5.32 to conclude that every group topology τ on G such that ζN2 ≤ τ ≤ ζN1 is almost trivial. Moreover these τ are finitely many because [N2 : N1] is finite. � Remark 5.34. A group G is hereditarily non-topologizable in case every sub- group of G is totally Markov [18]. Thus hereditarily non-topologizable ⇒ totally Markov ⇒ Markov. Consequently every group topology on a hereditarily non-topologizable group is almost trivial. If a hereditarily non-topologizable group G is Arnautov, then every quotient of G is Arnautov. While infinite Arnautov groups exist (see Example 5.29(a)), it is not known if there exists any infinite non-topologizable group. The existence of such a group would solve an open problem from [10]. 6. Problem B We start by underlying an important aspect of AK-completeness compared to K-minimality, where K is a class of topological groups. Indeed, let us re- call first that AG-completeness coincides with A-completeness and implies AK- completeness (see Remark 1.10). The K-minimal groups are precisely the in- discrete groups, whenever K contains all indiscrete groups. This fails to be true for AK-completeness. In fact, the group G = S(Z), equipped with either the discrete or the pointwise convergence topology, is A-complete (so AK-complete, for every K ⊆ G) as shown by Example 5.18(a,b). More generally for every non- trivial G ∈ T, the (obviously) non-indiscrete group (G, δG) is A-complete (so AK-complete, for every K ⊆ G) by Theorem 5.13. As we have seen in Section 5 A-complete (i.e., AG-complete) groups are not easy to come by. In order to have a richer choice of groups, we consider AK- complete groups for appropriate subclasses K of G. In case the subclass K is completely determined by an algebraic property (i.e., for every group topology τ on G, (G, τ ) ∈ K if and only if (G, δG) ∈ K), then obviously a topological group (G, τ ) ∈ K is AK-complete if and only if it is A-complete. A typical example to this effect is the class of all topological abelian groups, or more generally the class of all topological groups such that the underlying group belongs to a 112 D. Dikranjan and A. Giordano Bruno fixed variety V (in the sense of [20]) of abstract groups. We formulate an open question for a specific V in Question 6.13. In the sequel we consider subclasses K ⊆ G of a different form, most often K ⊆ H. Since H-minimality coincides with minimality, AH-completeness is a gener- alization of minimality. It is a strict generalization in view of (a) of the next example. Example 6.1. (a) The group (S(Z), δS(Z)) is A-complete, as shown by Example 5.18(b), and consequently AH-complete, but it is not minimal: δS(Z) and the point-wise convergence topology T are both Hausdorff. (b) Let G ∈ T be non-torsion. Then (G, δG) is A-complete by Theorem 5.13, and in particular it is AH-complete. On the other hand, by our hypothesis there exists x ∈ G of infinite order, that is 〈x〉 is abelian and so not AH-complete. This shows that in general a subgroup of an AH-complete group need not be AH-complete. (This is noted after Example 5.18 for the particular case of (S(Z), δS(Z)).) Anyway AH-completeness coincides with minimality in the abelian case: Proposition 6.2. If G is an abelian group and (G, τ ) ∈ H, then (G, τ ) is AH-complete if and only if it is minimal. This proposition gives a partial answer to Problem B for the subclass of H of abelian topological groups. The problem remains open for the larger class H: Question 6.3. When is a topological group (G, τ ) ∈ H AH-complete? And in which cases is (G, δG) AH-complete? The next example, that extends Example 5.18(a), motivates Lemma 6.5. Example 6.4. For an infinite topologically simple (i.e., there exists no non- trivial closed normal subgroup) Hausdorff non-abelian group (G, τ ), minimal implies A-complete. In fact Z(G) = {eG} and Proposition 5.9 applies. The next lemma and corollary provide partial answers to Question 6.3. Lemma 6.5 in particular covers the previous example, since it implies that every minimal group with trivial center is A-complete (in view of the fact that minimal implies AH-minimal). Lemma 6.5. Let G be a group with Z(G) = {eG} and let τ be a Hausdorff group topology on G. Then (G, τ ) is AH-complete if and only if (G, τ ) is A- complete. Proof. If (G, τ ) is A-complete, then it is AH-complete. Suppose that (G, τ ) is AH-complete. Let σ ≤ τ be a group topology on G such that idG : (G, τ ) → (G, σ) is semitopological. By Proposition 2.11 σ is Hausdorff. Then σ = τ . This proves that (G, τ ) is A-complete. � Arnautov’s problems on semitopological isomorphisms 113 This lemma implies Proposition 5.9, since minimal groups are AH-complete. Corollary 6.6. Let G be a group. Then Z(G) = {eG} and δG is AH-complete if and only if G ∈ T. Proof. If Z(G) = {eG} and δG is AH-complete, then δG is A-complete by Lemma 6.5 and so G ∈ T by Theorem 5.13. Assume that G ∈ T. By Theorem 5.13 δG is A-complete and so AH- complete. Moreover Z(G) = {eG} by Proposition 4.5(a). � Lemma 6.5 suggests the following question: is Z(G) = {eG} a necessary condition for the validity of the implication (G, τ ) AH-complete ⇒ (G, τ ) A- complete? According to Corollary 6.6 the answer is “yes” in case τ is the discrete topology. Proposition 6.7. Let (G, τ ) be a SIN Hausdorff group. If (G, τ ) is A-complete, then Z(G) = {eG}. Proof. Suppose that Z(G) 6= {eG}. We want to see that (G, τ ) fails to be A-complete. Consider the topology T := τ ∧ ζZ(G), which has as a local base at eG the family BT = {U · Z(G) : U ∈ V(G,τ )(eG)}. Since τ is Hausdorff and T is not Hausdorff (because Z(G) 6= {eG}), τ > T . So it remains to prove that idG : (G, τ ) → (G, T ) is semitopological. Since (G, τ ) is SIN, it suffices to prove that for every U ∈ V(G,τ )(eG) and for a fixed g ∈ G there exists Vg ∈ BT such that [g, Vg] ⊆ U and then apply Proposition 2.8. So let U ∈ V(G,τ )(eG) and g ∈ G. Since (G, τ ) is SIN, there exists U ′ ∈ V(G,τ )(eG) such that U ′U ′ ⊆ U and gU ′g−1 ⊆ U ′. Let Vg = U ′·Z(G) ∈ BT . Then [g, Vg] = [g, U ′] ⊆ U ′U ′ ⊆ U . Since we have proved that idG : (G, τ ) → (G, T ) is semitopological and τ > T , then (G, τ ) fails to be A-complete. � Remark 6.8. As a consequence of Lemma 6.5 and Proposition 6.7 we have the following equivalence between A-completeness and the purely algebraic property of having trivial center. Indeed, if (G, τ ) ∈ H is AH-complete, then Z(G) = {eH} implies (G, τ ) A-complete by Lemma 6.5. Moreover, if (G, τ ) is SIN, in view of Proposition 6.7 also the converse implication holds, that is, (G, τ ) is A-complete if and only if Z(G) = {eG}. Corollary 6.9. Let (G, τ ) be a Hausdorff group with Z(G) 6= {eG}. (a) If (G, τ ) is SIN and AH-complete, then it is not A-complete. (b) If (G, τ ) is SIN and minimal, then it is not A-complete. (c) If (G, τ ) is compact, then it is not A-complete. This corollary produces in particular examples of AH-complete groups which are not A-complete (e.g., compact groups with non-trivial center), showing that the implication (G, τ ) AH-complete ⇒ (G, τ ) A-complete may fail to be true, also for non-discrete groups. In particular in Example 6.12 shows a group, with non-trivial center, which does not admit any compact topology, but admits minimal linear (so SIN) topologies, that are not A-complete by Corollary 6.9. 114 D. Dikranjan and A. Giordano Bruno Proposition 6.10. Let G be a group such that G ∈ T and let F be a finite group. Then δG×F is AH-complete. Proof. Let τ be a Hausdorff group topology on G×F and suppose that idG×F : (G × F, δG×F ) → (G × F, τ ) is semitopological. By Remark 5.12 τ ≥ TG×F . But TG×F = TG × TF = δG × TF by Lemma 4.4. So τ ≥ δG × TF . Since τ is Hausdorff, τ = δG×F , and this proves that δG×F is AH-complete. � Using this proposition we can give examples of AH-complete groups which are not A-complete, as the following. Another example of an AH-complete group which is not A-complete is in Example 6.12. Example 6.11. Let G = S(Z) × Z(2). By Theorem 4.18(a) S(Z) ∈ T. Then (G, δG) is AH-complete by Proposition 6.10. Since Z(G) = {idZ} × Z(2) is not trivial, G 6∈ T by Proposition 4.5(a). Consequently G is not A-complete by Theorem 5.13. Example 6.12. Let p ∈ P and let G be the group HZ (see Example 4.3) equipped with the product topology T = P (τp, τp, τp) where τp is the p-adic topology of Z. A base of T is given by the family of the (normal) subgroups formed by the matrices of the form   1 pnZ pnZ 0 1 pnZ 0 0 1   . Clearly G is SIN. Then (G, T ) is minimal [5, 7], so AH-complete. Moreover (G, T ) is A-complete by Corollary 6.9. Considering SIN groups in Example 5.4, Proposition 6.7 and Corollary 6.9 we have weakened the commutativity from a topological point of view. A dif- ferent way to weaken commutativity, but algebraically, is to consider nilpotent topological groups: Question 6.13. If (G, τ ) is a nilpotent topological group, when is (G, τ ) A- complete? The following example is dedicated to a very particular case of this question. Example 6.14. Consider the class K0 R := {(HR, P (τ, τ, τ )) : τ is a ring topology on R}, where P (τ, τ, τ ) denotes the product topology on G. Then every G ∈ K0 R is AK0 R -complete. Indeed, let τ ≥ σ be ring topologies on R such that (HR, P (τ, τ, τ )), (HR, P (σ, σ, σ)) ∈ K 0 R . Suppose that idR : (HR, P (τ, τ, τ )) → (HR, P (σ, σ, σ)) is semitopological. By Theorem 1.2, for every U ′ =   1 U U 0 1 U 0 0 1   ∈ V(HR,P (τ,τ,τ ))(eHR ) and h = Arnautov’s problems on semitopological isomorphisms 115   1 1 0 0 1 0 0 0 1   there exists Vh =   1 V V 0 1 V 0 0 1   ∈ V(HR ,P (σ,σ,σ))(eHR ) such that [h, Vh] ⊆ U ′. In particular this implies V ⊆ U and hence σ ≥ τ , that is σ = τ . In a forthcoming paper [6] we extend this result to the more general case of generalized Heisenberg groups on an arbitrary unitary ring A. 7. Problem C Problem C is about compositions of semitopological isomorphisms. In order to measure more precisely the level of being semitopological, we introduce the next notion. Definition 7.1. Let G be a group, σ ≤ τ group topologies on G and n ∈ N+. Then idG : (G, τ ) → (G, σ) is n-step semitopological if there exist n − 1 group topologies σ ≤ λn−1 ≤ . . . ≤ λ1 ≤ τ on G such that idG : (G, τ ) → (G, λ1), idG : (G, λ1) → (G, λ2), . . . , idG : (G, λn−1) → (G, σ) are semitopological. Obviously idG : (G, τ ) → (G, σ) is 1-step semitopological if and only if it is semitopological. Moreover a continuous isomorphism of topological groups is composition of semitopological isomorphisms if and only if it is n-step semi- topological for some n ∈ N+. Let G be a non-trivial group. The lower central series of G is defined by γ1(G) = G and γn(G) = [G, γn−1(G)] for every n ∈ N, n ≥ 2. The upper central series of G is defined by Z0(G) = {eG}, Z1(G) = Z(G) and Zn(G) is such that Zn(G)/Zn−1(G) = Z(G/Zn−1(G)) for every n ∈ N, n ≥ 2. A group G is nilpotent if and only if γn(G) = {eG} for some n ∈ N+, if and only if Zm(G) = G for some m ∈ N+. The minimum n ∈ N+ such that γn+1(G) = {eG}, that is, the minimum n ∈ N+ such that Zn(G) = G, is the class of nilpotency of G. Our main theorem about n-step semitopological isomorphisms is the follow- ing. It is an answer to Problem C(a) in the particular case when the topologies on the domain and on the codomain are the discrete and the indiscrete one respectively. Theorem 7.2. Let G be a group and n ∈ N+. Then idG : (G, δG) → (G, ιG) is n-step semitopological if and only if G is nilpotent of class ≤ n. Proof. If idG : (G, δG) → (G, ιG) is n-step semitopological, then there exist n − 1 group topologies λn−1 ≤ . . . ≤ λ1 on G such that idG : (G, δG) → (G, λ1), idG : (G, λ1) → (G, λ2), . . . . . . , idG : (G, λn−2) → (G, λn−1), idG : (G, λn−1) → (G, ιG) are semitopological. By Theorem 2.7(b) G′ ⊆ V for every V ∈ V(G,λn−1)(eG). Since idG : (G, λn−2) → (G, λn−1) is semitopological, Theorem 1.2 implies that for every U ∈ V(G,λn−2)(eG) and for every g ∈ G there exists Vg ∈ V(G,λn−1)(eG) such that [g, Vg] ⊆ U . Consequently [g, G ′] ⊆ U for every U ∈ V(G,λn−2)(eG). 116 D. Dikranjan and A. Giordano Bruno Hence γ3(G) = [G, G ′] ⊆ U for every U ∈ V(G,λn−2)(eG). Proceeding by induction we have that γn(G) ⊆ U for every U ∈ V(G,λ1)(eG). By Theorem 2.7(a) cG(g) is λ1-open for every g ∈ G. Thus γn(G) ⊆ Z(G) and this implies that G is nilpotent of class ≤ n (γn+1(G) = {eG}). Conversely, if G is nilpotent of class ≤ n, consider on G the group topologies ζZ(G), ζZ2(G), . . . , ζZn−1(G). Then idG : (G, δG) → (G, ζZ(G)) is semitopological by Theorem 2.7(a) and idG : (G, ζZn−1(G)) → (G, ιG) is semitopological because G′ ≤ Zn−1(G) since G/Zn−1(G) is abelian and applying Theorem 2.7(b). For every i = 1, . . . , n − 1, by Corollary 3.13 idG : (G, ζZi (G)) → (G, ζZi+1(G)) is semitopological if and only if [G, Zi+1(G)] ≤ Zi(G) and this holds true since Zi+1(G)/Zi(G) = Z(G/Zi(G)). � As a particular case of n = 2 in this theorem, we find [2, Example 12], which witnesses that the composition of semitopological isomorphisms is not semi- topological in general. Indeed idG : (G, δG) → (G, ιG) is not semitopological, whenever G is not abelian. For n ∈ N+, let n-S := {fn ◦ . . . ◦ f1 : fi ∈ S}. Observe that S = 1-S ⊂ 2-S ⊂ . . . ⊂ n-S ⊂ (n + 1)-S ⊂ . . . , where all inclusions are proper by the previous theorem. Define also ∞-S := ⋃∞ n=1 n-S and observe that it is closed under compo- sitions. Moreover ∞-S is closed also under taking subgroups, quotients and finite products, in the following sense: Lemma 7.3. Let n ∈ N+, let G be a group and τ ≥ σ group topologies on G such that idG : (G, τ ) → (G, σ) is n-step semitopological. (a) If A is a subgroup of G, then idG ↾A= idA : A → A is n-step semitopo- logical. (b) If A is a normal subgroup of G, then idG/A : (G/A, τq) → (G/A, σq ) is n-step semitopological. Proof. (a) By hypothesis there exist n − 1 group topologies σ ≤ λn−1 ≤ . . . ≤ λ1 ≤ τ on G such that idG : (G, τ ) → (G, λ1), idG : (G, λ1) → (G, λ2), . . . , idG : (G, λn−1) → (G, σ) are semitopological. Theorem 2.3(a) implies that idA : (A, τ ↾A) → (A, λ1 ↾A), idA : (A, λ1 ↾A) → (A, λ2 ↾A), . . . . . . , idA : (A, λn−1 ↾A) → (A, σ ↾A) are semitopological and so idA : (A, τ ↾A) → (A, σ ↾A) is n-step semitopologi- cal. (b) Follows from Theorem 2.3(b). � Arnautov’s problems on semitopological isomorphisms 117 The following lemma shows that for each n ∈ N+ the class n-S is closed under taking products. In particular it implies that ∞-S is closed under taking finite products. Lemma 7.4. Let n ∈ N+, let {Gi : i ∈ I} be a family of groups and {τi : i ∈ I}, {σi : i ∈ I} two families of group topologies such that σi ≤ τi are group topologies on Gi and idGi : (Gi, τi) → (Gi, σi) is n-step semitopological for every i ∈ I. Then ∏ i∈I idGi : ∏ i∈I (Gi, τi) → ∏ i∈I (Gi, σi) is n-step semitopological. Proof. It follows from Theorem 2.4. � The following example shows that ∞-S is not closed under taking infinite direct products and answers negatively (b) of Problem C. In fact we construct a continuous isomorphism which is not composition of semitopological isomor- phisms. Example 7.5. For every n ∈ N+ let Gn be a nilpotent group of class n. Then ∏∞ n=1 idGn : ∏∞ n=1(Gn, δGn ) → ∏∞ n=1(Gn, ιGn ) is n-step semitopological for no n ∈ N+. Indeed idGn+1 : (Gn+1, δGn+1 ) → (Gn+1, ιGn+1 ) is not n-step semitopological whenever n ∈ N+, in view of Theorem 7.2, because Gn+1 is not nilpotent of class ≤ n. The next example is another particular case in which we answer Problem C(a). Example 7.6. Let n ∈ N+, let G be a totally Markov group and τ, σ group topologies on G. Every group topology on G is almost trivial by Proposition 5.21. Then idG : (G, τ ) → (G, σ) is n-step semitopological if and only if [G, [G, [...[G ︸ ︷︷ ︸ n , Nσ]]]] ≤ Nτ . In fact, suppose that idG : (G, τ ) → (G, σ) is n-step semitopological. Then there exist group topologies σ ≤ λn−1, ≤ . . . , ≤ λ1 ≤ τ on G such that idG : (G, τ ) → (G, λ1), idG : (G, λ1) → (G, λ2), . . . . . . , idG : (G, λn−1) → (G, σ) are semitopological. By Corollary 3.13 [G, Nσ] ⊆ Nλ1 , [G, Nλ1 ] ⊆ Nλ2 , . . . , [G, Nλn−1 ] ⊆ Nτ and hence [G, [G, [...[G ︸ ︷︷ ︸ n , Nσ]]]] ≤ Nτ . Assume that [G, [G, [...[G ︸ ︷︷ ︸ n , Nσ]]]] ≤ Nτ . Let Nλ1 = [G, Nσ], Nλ2 = [G, Nλ1 ], . . . , Nλn−1 = [G, Nλn−2 ]. By Corollary 3.13 and our assumption idG : (G, τ ) → (G, λ1), idG : (G, λ1) → (G, λ2), . . . , idG : (G, λn−1) → (G, σ) are semitopological. 118 D. Dikranjan and A. Giordano Bruno References [1] V. I. Arnautov, Semitopological isomorphisms of topological rings (Russian), Mathe- matical Investigations (1969) 4:2 (12), 3–16. [2] V. I. Arnautov, Semitopological isomorphisms of topological groups, Bul. Acad. Ştiinţe Repub. Mold. Mat. 2004 (2004), no. 1, 15–25. [3] S. Banach, Ueber metrische Gruppen, Studia Math. 3 (1931), 101–113. [4] L. 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J. 19 no. 5 (1978), 848–850 (1979).) [29] M. G. Tkachenko, Completeness of topological groups (Russian), Sibirsk. Mat. Zh. 25 (1984), no. 1, 146–158. Arnautov’s problems on semitopological isomorphisms 119 [30] M. G. Tkachenko, Some properties of free topological groups (Russian), Mat. Zametki 37 (1985), no. 1, 110–118, 139. Received July 2008 Accepted October 2008 Dikran Dikranjan (dikran.dikranjan@dimi.uniud.it) Dipartimento di Matematica e Informatica, Università di Udine, via delle Scienze, 206 - 33100 Udine, Italy Anna Giordano Bruno (anna.giordanobruno@dimi.uniud.it) Dipartimento di Matematica e Informatica, Università di Udine, via delle Scienze, 206 - 33100 Udine, Italy