On embedding properties of SD groups

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PII. S0161171204211280 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON EMBEDDING PROPERTIES OF

SD

_-GROUPS

VAHAGN H. MIKAELIAN

Received 19 November 2002

To the memory of Professor Alexey Ivanovich Kostrikin

Continuing our recent research on embedding properties of generalized soluble and gener-alized nilpotent groups, we study some embedding properties ofSD-groups. We show that every countableSD-groupGcan be subnormally embedded into a two-generatorSD-group H. This embedding can have additional properties: if the groupGis fully ordered, then the group Hcan be chosen to be also fully ordered. For any nontrivial word setV, this em-bedding can be constructed so that the image ofGunder the embedding lies in the verbal subgroupV (H)ofH. The main argument of the proof is used to build continuum examples ofSD-groups which are not locally soluble.

2000 Mathematics Subject Classiﬁcation: 20F14, 20F19, 20E10, 20E15, 20F60.

1. Results and background information

1.1. Background information and the main result. In [7] Kovács and Neumann con-sidered some embedding properties ofSN∗- andSI∗-groups (see deﬁnitions and ref-erences below). They, in particular, showed that every countableSN∗- orSI∗-group is embeddable into a two-generatorSN∗- or SI∗-group, respectively. The consideration of such embeddings of generalized soluble and generalized nilpotent groups was very natural after a series of results on embeddings into two-generator groups for soluble and nilpotent groups (see [1,18,19,20,23] and the references therein). And, in gen-eral, interest in embeddings of countable groups into two-generator groups (with some additional properties or conditions) is explained by the famous theorem of Higman, B. H. Neumann, and H. Neumann about the embeddability of an arbitrary countable group into a two-generator group [5].

In [13,15] we considered the embedding properties of a few other classes of gen-eralized soluble and gengen-eralized nilpotent groups. In particular, we saw that (a) every countableSN-,SI-,SN`-, orSI`-group is embeddable into a two-generatorSN-,SI-,SN`-, orSI`-group, respectively; but (b)not every countableZA- or N-group is embeddable into a two-generatorZA- orN-group, respectively.

The ﬁrst aim of the present paper is to consider similar problems for another popular class of generalized soluble groups: forSD-groups; that is, for groups in which the series of commutator subgroups,

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reaches unity:G(ρ)_{= {}₁_}_{for some ﬁnite or inﬁnite ordinal}_{ρ. As we will see, for any}

countableSD-group, such an embedding into a two-generatorSD-group is possible. Moreover—and this is the second aim of this paper—the mentioned embedding can satisfy a few additional properties: the embedding can besubnormal,verbal, andfully ordered. To have our statements in a precise form, we ﬁrst state our main theorem, and then turn to the background information about each of these three properties.

Theorem _1.1. _(A)_{Every countable} _SD-group _G_{is subnormally embeddable into a} two-generatorSD-groupH: there existsγ:G→Hsuch thatGγ(G),γ(G)H.

(B)For every nontrivial word setV⊆F_∞, the two-generatorSD-groupHand the em-beddingγcan be chosen so thatγ(G)lies in the verbal subgroupV (H)of the groupH. (C)Moreover, if the groupGis fully ordered, then the groupH can be chosen to be also fully ordered such thatGis order-isomorphic to its imageγ(G). Also, if the group Gis torsion-free, the groupHcan be chosen to be torsion-free.

1.2. The additional properties for embeddings. That the embedding of a general countable group into a two-generator group can be subnormal is proved by Dark in [1] (see also the paper of Hall [4]). In [14,15, 16] we combined the subnormality of the embeddings of countable groups with other properties (see below).

The consideration of the verbality of embeddings (of countable groups into two-generator groups) was initiated by B. H. Neumann and H. Neumann in [20], where they proved that every countable groupGcan be embedded not only into a two-generator groupH but also into the second commutator subgroupH(2)₌_H _{of the latter (the}

commutator subgroups are simply the special cases for verbal subgroups). In fact, here the second commutator subgroup can be replaced by any verbal subgroupV (H)[14]. If the groupGis anSN-,SI-,SN`-, orSI`-group, then the groupH can be constructed to belong to the same class asG[15]. And asTheorem 1.1shows now, the analog of this fact is also true for the class ofSD-groups. Moreover, all these embeddings can be subnormal.

The problem whether a fully ordered countable group can be embedded into a fully ordered two-generator group was posed by Neumann, and he solved it in [18]. In [16] this property was combined with subnormality and verbality of embeddings. Moreover, if the fully ordered countable groupGis anSN-,SI-,SN`-, orSI`-group, then the fully ordered groupHcan be chosen in the same class [15]. Again,Theorem 1.1shows that the analog of this is true for the class ofSD-groups.

1.3. An application of the argument. In [13] we used the embedding construction of [15] to build sets consisting of continuum of not locally soluble SI∗-groups. The reason why we devoted a paper to that topic was that so far in the literature there was only one example of the mentioned type. To be exact, there were two examples (independently built by Hall [3] and by Kovács and Neumann [7]) presenting thesame group. The examples built in [13] were not only pairwise distinct groups but, moreover, groups generating pairwise distinct varieties of groups.

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[13] as an illustration of what can be obtained by means of the verbal embeddings of groups.

Theorem1.2. There exists a continuum of torsion-free, not locally soluble two-gene-ratorSD-groups which generate pairwise diﬀerent varieties of groups.

1.4. References to the basic literature. AnSN∗-group or anSI∗-group is a group possessing a soluble ascending subnormal or normal series, respectively. In analogy with this, anSN_{`- or}SI_{`-group is a group possessing a soluble descending subnormal or} normal series respectively [12]. More generally, anSN- orSI-group is a group possess-ing a soluble subnormal or normal (not necessarily well ordered) system, respectively. AZA-group is a group with central ascending series. Finally, anN-group is a group in which every subgroup can be included in an ascending subnormal series. For informa-tion on the theory of generalized soluble and generalized nilpotent groups, we refer to the original articles of Kurosh and Chernikov [9] and of Plotkin [24,25], as well as to the books of Robinson [26,27] and of Kurosh [8]. For general information on varieties of groups, we refer to the basic book of Neumann [21]. Information on linearly (or fully) ordered groups can be found in the book of Fuchs [2] or in the papers of Levi [10,11] and of Neumann [17,18].

2. The main embedding constructions

2.1. The subnormal embedding into a two-generator group. We begin with a simple but still very useful construction (see, e.g., [3,20]). Assume the given groupGto be countable and its elements to be indexed by the set of nonnegative integers:

G=g0, g1, . . . , gn, . . .

. (2.1)

Then consider in the Cartesian wreath product GWrf of the group G and of the inﬁnite cyclic group generated by the elementf the following elementωof the base subgroupGf _:

ωfi₌   

gk, ifi=2k, k=0,1,2, . . . ,

1, ifi∈Z\2k_|_k₌_{0,1,2, . . .}_. (2.2)

Clearly, the elementf acts on the base subgroup as a “right-shift” operator. In par-ticular, for arbitrarygn, we haveωf−

2n

(1)=gn. Thus, for each pairgn, gm∈G, we

have

ωf−2n_{, ω}f−2m ₍₁₎₌_g n, gm

. (2.3)

Furthermore, for arbitraryj=0,

ωf−2n, ωf−2m fj=1. (2.4)

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Lemma2.1. LetH=H(G)be the above-constructed two-generator group. Then there is an isomorphic embeddingα:G→Hof the commutator subgroupGintoHsuch that α(G)is subnormal inH. Moreover, if the groupGhas any one of the properties

(1) Gis anSD-group,

(2) Gis a fully ordered group, (3) Gis a torsion-free group,

then the groupHalso has the same properties (and if the groupGis fully ordered, then Gis order-isomorphic toα(G)).

Proof. For understandable reasons, we can omit the case whenGis a trivial group.

For any pairgn, gm∈G, the embeddingαcan be deﬁned as

αgn, gm

=ωf−2n, ωf−2m ∈H. (2.5)

Equalities (2.3) and (2.4) show that this can be continued to an injection.

(1) Assume thatGis anSD-group:G(δ)_{= {}₁_}_{. Evidently,}_H_{lies in the normal closure}

T= ω H_of_ω_in_{H. As it is known (and can be calculated based on equalities (}_2.3₎

and (2.4)), the commutatorT is equal to thedirect powerGf of the commutator G. The direct powerTis anSD-group of lengthδ−1. Thus (taking into account the fact that the subgroups ofSD-groups areSD-groups) we get thatHis anSD-group of length 2+δ−1=δ+1.

(2) Assume that the full-order relation “≺” is deﬁned onG, and “lift” it to the group H. It is easy to see that for any elementfi_τ_∈_{H, where} _fi_∈_f _and _τ_∈_H_∩_Gf _,

there necessarily exists a corresponding maximal indexz0(τ)∈Zsuch thatτ(fi₎₌₁

for anyi≤z0(τ). Thus we can compare any two distinct elements ofH,

fi1_τ1_≺_fi2_τ2, _(2.6)

if and only ifi1< i2, or ifi1=i2andτ1(fz0+1₎≺_τ2(fz0+1_{), where}_z0_{is the minimum} ofz0(τ1)andz0(τ2).

(3) If the groupGis torsion-free, then the groupHis also torsion-free.

2.2. The verbal subnormal embedding. LetAbe the variety of all abelian groups and letVbe any variety diﬀerent from the variety of all groupsO(for now the properties of this variety are immaterial, but later it will be replaced by special varieties).

Lemma_2.2. _{For any variety}V=O_{, there exists a group}_N_{with the following properties:} (1) Nis a torsion-freeSD-group;

(2) Ngenerates a product varietyV1A, whereV1=OandV1is not properly contained inV(in particular,N∉VA);

(3) Ncan be fully ordered.

Proof. There are many methods to construct such a group. We outline one of them.

Consider the relatively free nilpotent groupS=Fk(Nc)of some rankkand classc≤k,

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for the groupGinSection 2.1. Consider in the inﬁnite Cartesian powerSz , that is, in the base subgroup of the Cartesian wreath productSWrz (of the groupS and of the inﬁnite cyclic groupz ) an elementλs for eachs∈S:

λs

zi₌   

s, ifi≥0,

1, ifi <0, (2.7)

and deﬁneN= λs, z|s∈S . The groupNcontains the ﬁrst copyS0ofS inSz. In

the sequel, we identifyS withS0and use the same notations for their elements if no misunderstanding arises.

(1) The groupNis clearly a torsion-free soluble group.

(2) The groupN belongs toV1A=NcA, andN contains a subgroup isomorphic to

the direct wreath productSwrz . Thus, var(N)=V1A=NcA.

(3) Since the verbal subgroupV (S)is not trivial (Vis the word set corresponding to the varietyV), we can choose an elementa∈V (S)of infinite order. On the free nilpotent groupS, a full-order relation “≺” can be defined as in [10,11,17]. Moreover, since in any group the full order can be replaced by its converse full order “≺−1_{” (x}_≺−1_y_{if and} only ify≺x), we can, without loss of generality, assume that the elementais positive: 1≺a. Then, according to the definition of full order, all the powersa2_{, a}3_{, . . . , a}n_{, . . .}

will also be positive elements. (This elementawill be used later.) We “lift” the full-order relation “≺” ofSto the groupN. As in the proof ofLemma 2.1, for any elementzi_τ_∈_N,

wherezi_∈_z _and_τ_∈_N_∩_Sz _{, there necessarily exists a maximal index}_z0(τ)_∈_Z_such

thatτ(zi₎₌_{1 for any}_i_≤_{z0(τ). Thus, for any two distinct elements of}_{N, we can put}

zi1_τ1_≺_zi2_τ2 _(2.8)

if and only ifi1< i2or ifi1=i2andτ1(zz0+1₎≺_τ2(zz0+1_{), where}_z0_{is the minimum} ofz0(τ1)andz0(τ2).

Now take a groupG, a nontrivial word setV, and the groupNconstructed as above for the givenV. Consider the Cartesian wreath productGWrNand select the following elements,χg, in the base groupGN of this wreath product:

χg(n)=   

g, ifn=ai_,_{for some positive integer}_i_∈_N_,

1, ifn∉ai_|_i_∈_N_, (2.9)

whereais the element chosen above. Denote byK=K(G, V )the following subgroup ofGWrN:

K=χg, N|g∈G. (2.10)

Denote by U the word set corresponding to the variety VA. In these notations, the following lemma holds.

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groupGintoKsuch thatβ(G)is subnormal inKand lies in the verbal subgroupU (K). Moreover, if the groupGhas any one of the properties

(1) Gis anSD-group,

(2) Gis a fully ordered group, (3) Gis a torsion-free group,

then the groupKalso has the same properties (and if the groupGis fully ordered, then it is order-isomorphic toβ(G)).

Proof. _Let_π_g_{be the element of the ﬁrst copy of}_G_in_GN_{corresponding to}_g∈_G:

πg(n)=   

g, ifn=1,

1, ifn∈N\{1}. (2.11)

The embeddingβcan be deﬁned as

β(g)=πg, ∀g∈G. (2.12)

Then(χ−1

g )aχg=πgbecause it is easy to calculate that

χ−1

g a

χg (n)=         

1, ifn∈N\ai_|_i₌_{0,1,2, . . .}_,

g, ifn=1=a0_,

1, ifn=a, a2_{, a}3_{, . . . .}

(2.13)

Since

a∈V (S)⊆U (N)⊆U (K), (2.14)

we haveπg=a−1aχg∈U (K). Thus the mappingg→πgdeﬁnes an isomorphic

embed-ding ofGonto the ﬁrst copyβ(G)ofGinGN_{and in}_{U (K). Clearly,}_β(G)_{is subnormal}

inK(and, in fact, even inGWrN).

(1) IfG is anSD-group:G(ν)_{= {1}, then}_K(ν+c+1)_{= {1}, where} _c_{is the nilpotency}

class ofS (in fact,ccould be replaced by a smaller integer, but it is immaterial for our purposes) for, clearly,K⊆ GN_{, S}z _,_K(c+1)_⊆_GN_{, and}_(GN₎(ν)_≤_(G(ν)₎N_{= {}₁_}_{. Notice}

that we could use this argument in the proof ofLemma 2.1(1). However, there we used a somewhat diﬀerent argument to stress that in that case we deal with a direct (not Cartesian) product.

(2) Assume that a full-order relation “≺” is deﬁned onG. From the deﬁnition of the elementsχg (and of the operation of elements of N onχg), it is clear that for any

nontrivial elementnθ∈K, wheren∈N andθ∈K∩GN_{, there necessarily exists an}

elementn0(θ)∈Nwith the following property:

θ(n)=1, ∀n≺n0(θ), θn0(θ)=1. (2.15)

Now the full orders available on the groupsGandNcan be “continued” to the group K. Letn1θ1andn2θ2be any two distinct elements ofK. Then

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if and only ifn1≺n2or ifn1=n2andθ1(n0)≺θ2(n0), wheren0is the minimum of n0(θ1)andn0(θ2).

(3) It is easy to see that if the groupGis torsion-free, then the groupGWrNand its subgroupKalso are torsion-free.

2.3. Proof ofTheorem 1.1. Lemmas2.1and2.3already allow us to prove the state-ments ofTheorem 1.1.

Proof of statement (A). _{Assume that}_G=_G0 _{is a countable}_{SD-group and}V= E is the trivial variety consisting of the group{1}only. ByLemma 2.3, the group G can be subnormally embedded into an SD-group G1 such that β(G)⊆U (G1)=G₁, where this time the word setUcorresponds to the variety of the abelian groupsU= VA=A. It is easy to see that the groupG1 is also countable. Thus, byLemma 2.1, the commutator subgroupG₁can be subnormally embedded into a two-generator SD-groupG2=H(G1),α:G1→G2. The subnormal embedding we are looking for can be deﬁned as the compositionβ·α.

Proof of statement (B). Assume thatV ⊆F_∞is any nontrivial word set

corre-sponding to the varietyV. Again, byLemma 2.3, the groupGcan be subnormally em-bedded into anSD-groupG1=K(G, V )such thatβ(G)⊆U (G1), whereUcorresponds toVA. ByLemma 2.1, the commutator subgroupG₁can be subnormally embedded into some two-generatorSD-groupG2=H(G1),α:G1→G2.

If we now show that

β(G)⊆VG₁, (2.17)

the statement will be proved because

(a) α(β(G))is subnormal inG2(forβ(G)is subnormal inG1, and the latter is sub-normal inG2);

(b) α(β(G))lies inV (G2)(forα(β(G))⊆α(V (G1))⊆V (G2)). To prove (2.17), we ﬁrst notice that

S0⊆N⊆G₁, (2.18)

where underS0we understand the ﬁrst copy ofS inSz_{. Indeed, we should simply}

apply the argument of the proof ofLemma 2.3to see thatS0lies inN. We have

a∈VS0⊆VG₁. (2.19)

Therefore, for anyg∈G,

πg=β(g)=a−1aχg∈V

G1

, (2.20)

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Proof of statement (C). This statement follows directly from Lemmas2.1 and

2.3and from the fact that a subgroup—in this case the commutator subgroupG1—of anSD-group (of a fully ordered group) is anSD-group (a fully ordered group). Also, if Gis torsion-free, evidently, the groupsG1andG2also have that property.Theorem 1.1

is then proved.

3. Continuum two-generatorSD-groups

3.1. Bisections. We need a special set of groups constructed by Ol’shanskii in [22]. Namely, let{Ln|n∈N}be a countable set of ﬁnite groups with the following

proper-ties:

(1) Ln∈L, whereL=Ois a soluble variety of ﬁnite exponent;

(2) Ln∉var(L1, . . . , Ln−1, Ln+1, . . . )for an arbitraryn∈N.

In fact, as varietyL, one can take the varietyG5∩B8qr, n=1,2, . . ., where G5is the

variety of soluble groups of length at most 5;q, r are distinct primes; andB8qr is the

Burnside variety of groups of exponents dividing 8qr [22].

We deﬁne abisection(B) as follows. The setNof positive integers can be split as (B) N_∪_N₌_{N, where}_N_∩_N_{= ∅}_and_N_,N_{= ∅.}

Then denote the groupL(B)to be the direct product of the groupsLn,n∈N, the variety

V(B)to be the variety generated by the groupsLn,n∈N=N\N, and the word setV(B)

to be that corresponding toV(B).

3.2. Construction ofSD-groups with bisections (B). Take any torsion-free insoluble SD-groupGnot generating the varietyOand put

G(B)=F∞var(G)varL(B). (3.1)

It is easy to see thatG(B)is anSD-group. Also, it is a torsion-free group by the theorem of Kovács about torsion-free relatively free groups of product varieties [6]. SinceV(B)is clearly a nontrivial word set, we are in a position to applyTheorem 1.1to subnormally embedG(B)into an appropriate two-generatorSD-group

H(B)=HG(B), V(B). (3.2)

Since the set of all the bisections (B) is of continuum cardinality, to proveTheorem 1.2, it is suﬃcient to ﬁnd many continuum bisections which determine groupsH(B)generating pairwise distinct varieties of groups.

Consider another bisection,

(˜B) ˜N_∪_N˜₌_N_{, where ˜}_N_∩_N˜_{= ∅}_{and ˜}_N_,_N˜_{= ∅}_,

diﬀerent from (B) and consider the groupsG(˜B)andH(˜B)=H(G(˜B), V(˜B))corresponding to this bisection (˜B). Here the inequality (B)=(˜B), of course, simply means thatN₌_N˜_(or,

equivalently,N₌_N˜_{). Clearly, var(L(B))}₌_var(L

(˜B)). Moreover, we have the following lemma.

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Proof. We have

varG(B)=var(G)varL(B),

varG_(˜_B)=var(G)varL_(˜_B). (3.3)

Then, by [21, Theorem 23.23], if var(G(B))=var(G(˜B)), we have var(L(B))=var(L(˜B)). This contradicts the fact that (B)=(˜B) and the selection of the group set{Ln|n∈N}.

The next step of our construction is the groupK(B)=K(G(B), V(B))that we built for the groupG(B) and the word setV(B). The set of all possible word sets (as well as the set of all varieties of groups) is of continuum cardinality. In fact, the continuum of distinct bisections (B) already provide us with continuum of distinct word setsV(B). This, however, does not mean that building the groupsK(B)for continuum distinct word sets V(B), we will getcontinuumexamples of distinctSD-groupsH(B). The point is that the role of the word setV inK(and, thus, inH) is in the determination of the classcand of the rankkof the free nilpotent groupS=Fk(Nc), which we are using to build the

appropriate groupN. The set of such integers iscountable. This means that there exist many (in fact, continuum) word setsV(B)for which the same nilpotent groupS(B)should be chosen. This also means that the construction, that we have at our disposal at the current moment, does not yet allow us to build a continuum ofSD-groups using the fact about continuum set of wordsV(B)only.

But this observation also allows us to modify and shorten one of the segments of our proof. Namely, we restrict ourselves to such a set (of continuum cardinality) of word setsV(B)which correspond to a ﬁxed pair of integerscandk. It will be suﬃcient to prove that this set can already give rise to continuum two-generator (torsion-free) SD-groups.

Lemma_3.2. _{Assume that (B) and (}˜_{B) are two distinct bisections of the type mentioned:} S(B)=S(˜B). Ifvar(G(B))=var(G(˜B)), thenvar(K(B))=var(K(˜B)).

Proof. As we saw in the proof ofLemma 2.3, for the given bisection (B), the group

K(B)contains the ﬁrst copy ofG(B). That copy, together withN(B), generates the direct wreath productG(B)wrN(B). Since the groupN(B)discriminates the varietyNcA, the group

G(B)wrN(B)generates var(G(B))var(N(B))=var(G(B))NcA=var(K(B)). Thus, taking another

bisection (˜B), we would get var(K(˜B))=var(G(˜B))NcA(recall that according to the remark

proceeding this lemma, we can use the same varietyNcfor both bisections). The latter

is distinct from var(G(B))NcAwhenever the bisections are distinct.

The ﬁnal step of our argument is the construction of the two-generator groupH(B)= H(K(B)).

Lemma_3.3. _If_var(K_(B)₎=_var(K₍_B)_˜_{), then}_var(H_(B)₎=_var(H₍_˜_B)_).

Proof. _{The proof immediately follows from the fact that}

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We take any nonidentityw=w(x1, . . . , xn)for the variety var(K(B))Aand show thatw

can be falsiﬁed on some elements ofH(B), as well. This will prove the point becauseH(B) evidently belongs to var(K(B))A. Takec1, . . . , cn∈K(B)wrf such thatw(c1, . . . , cn)=1.

Clearly,ci=fmiρi, where ρi belongs to the base subgroupK(B)f ;i=1, . . . , n. Finitely many elementsρiin this direct wreath product have only ﬁnitely many nontrivial

“co-ordinates” ρi(fj), i=1, . . . , n,j ∈Z. This means that there is a big enough positive

integern∗ such that if we replace (trivial) “coordinates”ρi(fj),|j|> n∗ of eachρi,

by arbitrarily chosen values from the groupK(B), and denote these new strings byρ_i correspondingly, then we will still have

wfm1_ρ

1, . . . , fmnρn,

=1. (3.5)

Since all the powersfm1_{, . . . , f}mn_{already belong to}_{H(K(B)), the proof will be completed} if we show the following rather more general fact: for arbitrary positive integern0and arbitrary pregiven valuesdj∈K(B),j= −n0, . . . , n0, the groupH(K(B))contains such an

elementρ∈H(K(B))∩K_(B)f for whichρ(fj₎₌_d

j,j= −n0, . . . , n0.

Taking into account the “shifting” eﬀect of the elementf, it will be suﬃcient to show that for any pregivend∈K(B), there is an elementρd ∈H(K(B))∩K

f

(B) such that

ρd

fj=

    

d, ifj=0,

1, if −2n0≤j≤2n0, j=0. (3.6)

(Notice that we did not put any requirements onρ(fj₎_for_{j >}_2n0_{or for}_{j <}₋_2n0.)

The elementsρwill then be products of elements of typeρ_d (for variousd’s) and of their conjugates by powers off. It remains to construct elementsρd (for anydand

n0) by means of two generatorsω(B)andf. We have

ω(B)fi=

    

gk, ifi=2k, k=0,1,2, . . . ,

1, ifi∈Z\2k_|_k₌_{0,1,2, . . .}_, (3.7)

where this time the countable groupK(B)is presented asK(B)= {g0, g1, . . .}. The element dcan be presented as a productgi·gj for inﬁnitely many pairsgi, gj∈K(B). On the

other hand, the number of all possible pairsgi, gjwith a common upper bound on|i|

and|j|is clearly ﬁnite. Thus, there necessarily exists a pairgi, gjsuch thatd=gi·gj

and 2i_,2j_>_{2n0. Then}

ρ_d =ωf−2i_ωf−2j_. _(3.8)

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And none of these groups is locally soluble because it would be then a soluble group, and, thus, it could not contain the initial groupG(B)that was chosen to be insoluble.

Acknowledgments. The author is very much thankful to Prof. Alexander Yu.

Ol’shanskii, Moscow State University and Vanderbilt University, for an advice (included in the paper asLemma 3.3) that allowed him to shorten the initial proof inSection 3, as well as to the referee for very valuable comments to the manuscript.

References

[1] R. S. Dark,On subnormal embedding theorems for groups, J. London Math. Soc.43(1968), 387–390.

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Vahagn H. Mikaelian: Department of Computer Science, Yerevan State University, 375025 Yere-van, Armenia