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Arnautov's problems on semitopological isomorphisms

Semitopological isomorphisms of topological groups were introduced by Arnautov, who posed several questions related to compositions of semitopological isomorphisms and the groups G (we call them Arnautov groups) such that for every group topology T on G every semitopological isomorphism with domain (G,T) 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.

The following notion is motivated by the fact that it is not always possible to prove the existence of such G and f , asking G to be also a normal subgroup of G (see also [1] for topological rings). 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 −1 U 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. (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 subgroups, 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]: 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.

Main Results
The next definition reminds the Br-completeness (since we impose openness only on certain continuous isomorphisms, namely, the semitopological 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.
We denote by A the class of all Arnautov groups.
Taȋmanov [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 neighborhood 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: • totally Taȋmanov if G/N is Taȋmanov for every N ⊳ G.
Obviously every simple Taȋmanov group is totally Taȋmanov. We denote by T and Tt the classes of Taȋmanov and totally Taȋmanov 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 don't know if they coincide (see Theorem 5.16 and Question 5.17). Example 5.18 considers properties of S(Z) related to Problem A. First of all it shows that Acompleteness 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 Taȋmanov and consequently not Arnautov. Nevertheless S(Z)/Sω(Z) is totally Taȋmanov 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): 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).
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 A K ′ -complete implies AKcomplete. 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-minimality 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 conjugation). 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 Taȋmanov 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 that is the group of upper unitriangular 3 × 3 matrices over R, endowed with different group topologies. The group H R 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 Theorem 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 algebraic. 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+.

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 −1 y −1 and for , 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 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 projection, 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.

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, σ) 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 Moreover the next proposition shows that semitopological is a "local" property, 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 : The following theorems show the stability for subgroups, quotients and products.
In a particular case, that is for initial topologies, the converse implication of Theorem 2.3(b) holds true: In the next theorem we consider the particular cases when one of the two topologies on G is trivial: (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 : 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: 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 .
In particular idG : (G, δG) → (G, ιG) is semitopological if and only if the group G is abelian.

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. 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 {e G } and ιG = ζG. This justifies the term used in Definition 3.1.
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 = sup α ζN α . 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 = sup α ζN α .
If (G, τ ) is a topological group, letτ denote the quotient topology of (G, τ ) with respect to the normal subgroup Nτ , which is indiscrete. Thenτ is Hausdorff. Moreover (G, τ ) is almost trivial if and only if (G/Nτ ,τ ) 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: 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 .

Permanence properties of the almost trivial topologies
The assignment N → 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 complete 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 {ζN i : i ∈ I} in AT (G) is simply ζ i∈I N i , whereas the meet of a family {ζN i : i ∈ I} in T (G) is the group topology having as prebase of the neighborhoods at eG the family {ζN i : i ∈ I} (in 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.
(b) For every normal subgroup N0 of G the quotient topology of ζN on G/N0 is almost trivial and coincides with ζ N 0 N/N 0 .
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 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. 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 = Nτ q . 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).
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.

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 : 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- 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. The next example is a consequence of this corollary.
Example 3.14. Let G be a group and ζN an almost trivial group topology on G. Consider Then: On a group G it is possible to consider the almost trivial topology generated by 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 : 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, ζN τ ) → (G, τ ) is continuous. Moreover, if ζL is another almost trivial topology on G such that idG : (G, ζL) → (G, τ ) is continuous, then idG : is a family of subgroups of G closed under finite intersections. Then the Taȋmanov topology TG has C as local base at eG, that is BT G = C.
We collect in the next lemma the first properties of this topology. (d) in case G is finitely generated, TG is almost trivial; in particular TG = δG if and only if Z(G) is trivial.

Permanence properties of the class T
The following results show that the Taȋmanov topology has nice properties. The next proposition proves that it is a functorial topology with respect to continuous surjective homomorphisms. On the other hand, the next example shows that the Taȋmanov topology is not functorial with respect to open surjective homomorphisms.
It follows from (a) that every non-trivial abelian group G ∈ 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.
Proof. Let G1, G2 ∈ T and G := G1 × G2. By Lemma 4.4 TG = TG 1 × TG 2 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. but can be also simply directly proved.
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 Taȋmanov (as shown in Example 4.7(c)), • there exist arbitrarily large Taȋmanov groups; moreover, every non-abelian subgroup of a nonabelian free group is Taȋmanov, being free [23]; • every group is quotient of a Taȋmanov group, since every group is quotient of a non-abelian free group of rank > 1 [23].
It is not clear if this holds also for subgroups:  Proof. Let f : G → G be defined by f (x) = −x for every x ∈ G. Moreover let 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 containing G as a subgroup and such that |H| = ω · |G|.
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 = 0. Then there exists H ∈ T such that G ≤ H and |H| = ω · |G|.
Proof of the claim. Let H := G ⋊ f .
Lemma 4.13 shows that to prove that a group is Taȋmanov it suffices to consider a convenient quotient with a finite normal subgroup and check whether it is Taȋmanov.  Proof. Let F1 be a finite subset of G/N such that c G/N (F1) = {e G/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 )) ⊆ c G/N (F1) = {e G/N }, cG(F ) ⊆ N . Since N is finite, Claim 4.12 applies to conclude that G ∈ T. The next is an example of a totally Taȋmanov group.
Example 4.14. We denote by G := SO3(R) the group of all orthogonal matrices 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 = β and Tα = T β , Tα ∩ T β is finite. Then cG(α) ∩ cG(β) is finite. By Claim 4.12 G ∈ T. Moreover H induces a partition of X, that is X = x∈R H OH (x), where RH ⊆ X is a set of representing elements.
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- 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 = τ ∈ c S(X) (H).
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. (c)⇒(b) Since κ > ω and H is countable, we can suppose that S has the property that Sα and S β are not conjugated in H for every α, β < κ with α = 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.
Define Xα := {hSα : h ∈ H} for every α < κ and X := α<κ Xα. Moreover let xα := idHSα ∈ 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 H := ϕ(H) ≤ S(X). Then H is finitely generated and the action of H on X is the same as the action of H on X.
(b) Let H be a finitely generated subgroup of S(X). Since OH (x) is countable 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) ∈ T. So Question 4.19 becomes: does there exists a finitely generated group H with a "large" (i.e., of cardinality κ with ω < κ ≤ c) family of self-normalizing subgroups?

Problem A
We start considering a stability property of the the class A of Arnautov groups. Proof. Let G ∈ A and let N be a normal subgroup of G. Let σ ≤ τ be group topologies on G/N such that id G/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 G ab = G/G ′ be the abelianization of G and endow G ab with the discrete topology and with the indiscrete topology: 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] ⊆ U for every Vg ∈ V (G,σ) (eG) (this follows from Proposition 2.8).
(b) If (G, τ ) is Hausdorff and τ ≤ ζ G ′ (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.
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; 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.
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. 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, ζN ) is not semitopological if sup{τ, ζN } > ζN . 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 ≤ NT G = Z(G).

When the discrete topology is A-complete
Consequently the Taȋmanov 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 Taȋmanov topology is discrete, that is, the group is Taȋmanov. This also motivates Definition 1.8.
The next theorem solves a particular case of Problem A, that is, it characterizes the groups for which the discrete topology is A-complete. 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 Taȋmanov group which has an infinite non-abelian subgroup that is not Taȋmanov. Proof. Suppose that ζN is A-complete. Since ζN is the initial topology of δ G/N , it follows that δ G/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 δ G/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: 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 Proposition 5.25.
The next examples show that a group can admit two A-complete topologies that are one strictly finer than the other.  (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 ∈ A, then τ σ ∈ A and so π(σ) ∈ π(τ ) . Moreover τ ∈ 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 ∈ T by Proposition 4.5(a).
(e) By (d) ζA is not A-complete in view of Theorem 5.15, hence G ∈ 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 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.

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. 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 ∈ 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) (e G/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. Proof. That Mt is closed under extensions is a direct consequence of Propositions 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. 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: Proof. If G ∈ T, apply Theorem 5.13.
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 Taȋmanov, which is precisely the answer to Question 5.17 in the particular case of totally Markov groups. (a) If τ is a group topology on G, the following conditions are equivalent: The following conditions are equivalent: In Example 5.18 we have seen that S(Z) ∈ 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) ∈ M. (c) In general a totally Markov group need not be an Arnautov group, that is, Mt ⊆ A; for example G := M × Z(2) ∈ Mt but G ∈ A.
Item (b) of this example shows that A is not stable under taking subgroups.  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 ζN 2 ≤ τ ≤ ζN 1 is almost trivial. In particular, (a) if N2 ∈ Mt, then every group topology τ on G with τ ≥ ζN 2 is almost trivial; and (b) if G/N1 ∈ Mt, then every group topology τ on G with τ ≤ ζN 1 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 ζN 2 ≤ τ ≤ ζN 1 and they are all almost trivial.
Proof. Apply Corollary 5.32 to conclude that every group topology τ on G such that ζN 2 ≤ τ ≤ ζN 1 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 subgroup 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].

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 recall first that AG-completeness coincides with Acompleteness and implies AK-completeness (see Remark 1.10). The K-minimal groups are precisely the indiscrete 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 AKcomplete 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 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 generalization of minimality. It is a strict generalization in view of (a) of the next example. (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), δ S(Z) ).) Anyway AH-completeness coincides with minimality in the abelian case: 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. 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).
(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. 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.
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.  [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 different 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. where P (τ, τ, τ ) denotes the product topology on G. Then every Indeed, let τ ≥ σ be ring topologies on R such that (H R , P (τ, τ, τ )), (H R , P (σ, σ, σ)) ∈ K 0 R .
In a forthcoming paper [6] we extend this result to the more general case of generalized Heisenberg groups on an arbitrary unitary ring A.

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.
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 semitopological 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.
As a particular case of n = 2 in this theorem, we find [2, Example 12], which witnesses that the composition of semitopological isomorphism is not semitopological in general. Indeed idG : (G, δG) → (G, ιG) is not semitopological, whenever G is not abelian. 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 compositions. 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 semitopological. 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 idG i : (Gi, τi) → (Gi, σi) is n-step semitopological for every i ∈ I. Then i∈I idG i : 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 isomorphisms.
The next example is another particular case in which we answer Problem C(a).