On Kaluzhnin-Krasner’s embedding of groups
In this note, we consider a ’thrifty’ version of Kaluzhnin-Krasner’s embedding in wreath products and apply it to extensions by finite groups and to metabelian groups.
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irk-123456789-1527892019-06-13T01:25:38Z On Kaluzhnin-Krasner’s embedding of groups Olshanskii, A.Yu. In this note, we consider a ’thrifty’ version of Kaluzhnin-Krasner’s embedding in wreath products and apply it to extensions by finite groups and to metabelian groups. 2015 Article On Kaluzhnin-Krasner’s embedding of groups / A.Yu. Olshanskii // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 1. — С. 77-86. — Бібліогр.: 12 назв. — англ. 1726-3255 2010 MSC:20E22, 20F16, 20E07. http://dspace.nbuv.gov.ua/handle/123456789/152789 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this note, we consider a ’thrifty’ version of Kaluzhnin-Krasner’s embedding in wreath products and apply it to extensions by finite groups and to metabelian groups. |
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Olshanskii, A.Yu. On Kaluzhnin-Krasner’s embedding of groups Algebra and Discrete Mathematics |
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Olshanskii, A.Yu. |
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Olshanskii, A.Yu. |
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On Kaluzhnin-Krasner’s embedding of groups |
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On Kaluzhnin-Krasner’s embedding of groups |
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On Kaluzhnin-Krasner’s embedding of groups |
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On Kaluzhnin-Krasner’s embedding of groups |
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On Kaluzhnin-Krasner’s embedding of groups |
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on kaluzhnin-krasner’s embedding of groups |
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Інститут прикладної математики і механіки НАН України |
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On Kaluzhnin-Krasner’s embedding of groups / A.Yu. Olshanskii // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 1. — С. 77-86. — Бібліогр.: 12 назв. — англ. |
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Algebra and Discrete Mathematics |
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2025-07-14T04:17:01Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 19 (2015). Number 1, pp. 77–86
© Journal “Algebra and Discrete Mathematics”
On Kaluzhnin-Krasner’s embedding of groups
A. Yu. Olshanskii1
Abstract. In this note, we consider a ’thrifty’ version of
Kaluzhnin-Krasner’s embedding in wreath products and apply it to
extensions by finite groups and to metabelian groups.
1. Introduction
This note goes back to the pioneer paper of L. Kaluzhnin and
M. Krasner [6], where wreath products of groups were introduced and
studied. Later many other group theorists applied wreath products to
construct various counter-examples and to prove embedding theorems,
and now wreath products are among the main tools of Group Theory.
Here I pay attention to a feature of Kaluzhnin-Krasner’s works, that
probably has not been used in subsequent research papers. Herewith I
consider only standard wreath products of abstract groups (i.e., in terms
of [6], of permutation groups with regular actions).
Let A and B be groups and F a group of all functions f : B → A with
multiplication (f1f2)(x) = f1(x)f2(x) for x ∈ B. The group B acts on F
from the right by shift automorphisms: (f ◦ b)(x) = f(xb−1) for all f ∈ F ,
b, x ∈ B, and the associated with this action semidirect product B ⋉ F is
called the (complete) wreath product of the groups A and B, denoted by
A Wr B. Thus, every element of A Wr B has a unique presentation as bf
(b ∈ B, f ∈ F ) and the multiplication rule follows from the conjugation
formula
(b−1fb)(x) = f(xb−1) (1)
in A Wr B for any b, x ∈ B and f ∈ F .
1The author was supported in part by the NSF grants DMS-1161294 and by the
RFBR grant 15-01-05823.
2010 MSC: 20E22, 20F16, 20E07.
Key words and phrases: wreath product, embedding of group, metabelian group.
78 On Kaluzhnin-Krasner’s embedding of groups
Observe that any homomorphism A → Ā induces the homomorphism
A Wr B → Ā Wr B by the rule bf 7→ bf̄ , where f̄ ∈ F̄ is obtained by
replacing the values of f by their images in Ā.
Given an arbitrary group G with a normal subgroup A, one has a
canonical homomorphism π of G onto the factor group G/A = B. Let
b 7→ bs be any transversal B → G, i.e. π(bs) = b. Then the Kaluzhnin -
Krasner monomorphism φ of the (abstract) group G into A Wr B is given
by the formula (see [5], [10])
φ(g) = π(g)fg, where fg(x) = (xπ(g)−1)s g (xs)−1 (2)
Applying π to (xπ(g)−1)s g (xs)−1, one obtains 1, and so fg ∈ F .
To check that φ(g1g2) = φ(g1)φ(g2), one just exploits the formulas (1,2).
Finally, φ is injective since obviously we have ker φ 6 A, and by (2),
fg(x) = xsg(xs)−1 6= 1 if g ∈ A\1.
The above-defined form of the Kaluzhnin - Krasner embedding φ is
well known, and, up to conjugation, the image φ(G) does not depend on
the transversal s. However, the original paper [6] suggested a stronger form
of such an embedding. Namely, assume now that a subgroup C is normal
in the normal subgroup A of G, but C does not contain nontrivial normal
subgroups of G. Then consider the product κ of φ and the homomorphism
A Wr B → Ā Wr B induces by the canonical homomorphism A → Ā =
A/C with a 7→ ā for a ∈ A. Thus
κ(g) = π(g)fg, where fg(x) = (xπ(g)−1)s g (xs)−1 (3)
If g ∈ ker κ, then π(g) = 1 and so g ∈ A. Then formula (3) shows that
the values xsg(xs)−1 are trivial in A/C for any x ∈ B, i.e. xsg(xs)−1 ∈ C.
Taking x = 1, we have g ∈ C since C is normal in A. Thus, ker κ 6 C,
which implies ker κ = 1 because C contains no nontrivial normal in G
subgroups.
Therefore κ is also an embedding of G, and the following is a slightly
modified version of the old Kaluzhnin - Krasner statement:
Proposition 1.1. Let G ⊲ A ⊲ C be a subnormal series of a group G
with factors Ā = A/C and B = G/A, and let C contain no non-trivial
normal subgroups of G. Then there is an isomorphic embedding κ of the
group G in the wreath product Ā Wr B = B ⋉ F̄ , the embedding κ can be
defined by the formula (3), so κ(A) 6 F̄ and κ(G)F̄ = Ā Wr B.
The group Ā can be much smaller than A, and making use of this
below, we
A. Yu. Olshanski i 79
• apply the embedding κ to wreath products with finite active group B,
• embed finitely generated metabelian groups into Ā Wr B with ’small’
abelian Ā and B, and
• observe that (Z/pZ) WrZ and ZWrZ contain 2ℵ0 non-isomorphic
locally polycyclic subgroups.
2. Splittings of some group extensions
The first application gives a characterization of wreath products with
finite active group. If the normal subgroup A is abelian, then the statement
of Theorem 2.1 follows from Proposition I.8.3 [7].
Theorem 2.1. Assume that a normal subgroup A of a group G is a direct
product ×n
i=1Hi, where {Hi}16i6n is the set of all conjugate to H = H1
subgroups of G. Also assume that the normalizer NG(H) is equal to A.
Then A has a semidirect compliment B in G and G = H Wr B.
Proof. Note that n is the index of the normalizer of H in G, therefore
the group B = G/A has order n.
Define C = C1 =
∏
i6=1 Hi. The normal in A subgroup C is conjugate
in G to any Cj =
∏
i6=j Hi, as it follows from the assumptions. Therefore
∩n
j=1Cj = 1, and C does not contain nontrivial normal in G subgroups.
By Proposition 1.1, we have the embedding κ of G in the wreath
product (A/C) Wr B. If g ∈ A, then by formula (3), κ(g) = f̄g, where
f̄g(x) = xsg(xs)−1. We may further assume that xs = 1 for x = 1. Then
for g ∈ H, every value xsg(xs)−1 of fg belongs to some Hi 6 C if x 6= 1
and fg(1) = g. Therefore all the values of f̄g, except for one, are trivial,
and κ(H) = H(1), where H(1) is the subgroup of functions f̄ supported
by {1} only. Since H(1) ≃ A/C ≃ H1 = H, the subgroup H(1) can be
identified with H.
The conjugacy class of the subgroup H(1) in the wreath product
H Wr B consists of n subgroups H(b) = b−1H(1)b, where b ∈ B. Therefore
the set {H(b)}b∈B is the κ-image of the set {Hi}
n
i=1. Since F̄ = ×b∈BH(b),
we have κ(A) = F̄ and, by Proposition 1.1, A Wr B = κ(G)F̄ = κ(G).
Hence κ is an isomorphism, and the theorem is proved.
The following example shows that one cannot remove the finiteness
of the quotient B = G/A from the assumption of Theorem 2.1.
Example. Let G be a free metabelian group with two generators a
and b. The commutator subgroup A = [G, G] is the normal closure of
80 On Kaluzhnin-Krasner’s embedding of groups
one commutator [a, b]. Since the subgroup A is abelian, every subgroup
conjugate to H = 〈[a, b]〉 is of the form Hij = c−1
ij 〈[a, b]〉cij , where cij =
aibj , i, j ∈ Z. We explain below the known fact that A is the direct product
×i,jHi,j . It follows that NG(H) = A. However G does not split over A
because by A. Shmelkin’s [12] result (also see [10], Theorem 42.56) two
independent modulo A elements must generate free metabelian subgroup
of G.
To check that the elements dij = c−1
ij 〈[a, b]〉cij are linearly independent
over Z, one can apply the homomorphism µ of G (in fact, a version of
Magnus’s embedding [8,8]) into the metabelian group of upper triangle
2 × 2 matrices over the group ring Z〈x, y〉 of a free abelian group of rank
two given by the rule
µ(a) = diag(x, 1), µ(b) = diag(y, 1) + E12,
where E12 is the matrix with 1 at position (1, 2) and zeros everywhere else.
The matrix multiplication shows that µ(dij) = I + x−iy−j−1(1 − x−1)E12,
where the elements x−iy−j−1(1 − x−1) of Z〈x, y〉 (i, j ∈ Z) are linearly
independent over Z.
3. Embeddings of metabelian groups
There are finitely generated torsion free metabelian groups which are
not embeddable in W = ZWr B with finitely generated abelian B. For
example, the derived subgroup [G, G] of the Baumslag - Solitar group
G = 〈a, b | b−1ab = an〉 is isomorphic to the additive group of rationals
whose denominators divide some powers of n, denote it by Dn; but for
|n| > 2, [W, W ] has no nontrivial elements divisible by all powers of n.
(Moreover, it is easy to construct 2-generated metabelian groups G with
[G, G] containing an infinite direct power of the group Dn [9].) However
the following is true.
Theorem 3.1. Let G be a finitely generated metabelian group with infinite
abelianization B = G/[G, G]. Then for some n = n(G) > 1,
(a) G embeds in the wreath product (Dn × Z/nZ) Wr B.
(b) G is isomorphic to a subgroup of W = Dn Wr B if the derived
subgroup [G, G] is torsion free;
(c) G is isomorphic to a subgroup of W = (Z/nZ) Wr B, provided the
derived subgroup [G, G] is a torsion group.
A. Yu. Olshanski i 81
The proof is based on the following
Lemma 3.2. Let G be a finitely generated metabelian group and A an
abelian normal subgroup of G. Then there is a subgroup C in A containing
no non-trivial normal in G subgroups, such that for some n > 1,
(a) the factor group A/C is isomorphic to a subgroup of a finite direct
power of the group Dn × (Z/nZ);
(b) A/C is isomorphic to a subgroup of a finite direct power of Dn if A
is torsion free;
(c) A/C is isomorphic to a subgroup of a finite direct power of Z/nZ
if A is a torsion group.
Proof. (b) We denote by R the maximal normal in G subgroup of finite
(torsion-free) rank contained in A. Since G satisfies the maximum con-
dition for normal subgroups [3], this ‘radical’ R exists and contains all
normal in G subgroups of A having finite rank. Moreover, the maximal
torsion subgroup T/R of L = A/R is trivial since the subgroup T is
normal in G and its rank is equal to the rank of R.
By Ph.Hall’s theorem on normal subgroups in metabelian groups
(see [4], lemmas 8 and 5.2), there is n > 1, and a basis (a1, a2, . . . ) of a
free abelian subgroup M 6 A such that A/M is a torsion group having
non-trivial p-subgroups for p|n only, and so the group A embeds in a
direct product D1
n × D2
n × . . . of the copies of Dn (such that ai 7→ Di
n).
We will use additive notation for A. If R = A, i.e. if the basis of M is
finite, the statement (b) holds for C = 0. So we assume further that M
has infinite rank.
Since R has finite rank, it follows from the above embedding of A into
the countable direct power of Dn that the group A has a subgroup K
such that the intersection R ∩ K is trivial and the factor group A/K is
isomorphic to a subgroup of a finite direct power of Dn.
Let us enumerate non-trivial normal subgroups N of G of infinite
rank, which are contained in A and have torsion free factor groups A/N :
N1, N2, . . . (This set is countable since G satisfies the maximum condition
for normal subgroups.) Using this enumeration, we will transform the
basis of M as follows.
Let a01 =a1, a02 = a2, . . . , and assume that the basis (ai−1,1, ai−1,2, . . . )
of M is defined for some i > 1, and ai−1,k = ak if k is greater than m(i−1),
where m(0) = 0.
82 On Kaluzhnin-Krasner’s embedding of groups
Since the subgroup Ni has infinite rank, Ni ∩ M has a non-zero
element gi =
∑
j>m(i−1) λjaj . Let m(i) be the maximal subscript at non-
zero coefficients λj of this sum. We may assume that the greatest common
divisor of the coefficients λm(i−1)+1, . . . , λm(i) is 1 since the group A/Ni
is torsion free. Therefore there exists a basis (ai,m(i−1)+1, . . . , ai,m(i)) of
the free abelian subgroup 〈am(i−1)+1, . . . , am(i)〉 with ai,m(i−1)+1 = gi.
The other elements of the (i − 1)-th basis of M are left unchanged, i.e.
ai,k = ai−1,k if k 6 m(i − 1) or k > m(i).
Now we define ek = ai,k if m(i−1) < k 6 (m(i)) for some i and obtain
a new basis of (e1, e2, . . . ) of M because we see from the construction that
〈e1, . . . , em(i)〉 = 〈a1, . . . , am(i)〉 for every m(i) > 0. Thus every subgroup
Ni contains an element gi = ai,m(i−1)+1 = em(i−1)+1 from this new basis.
Since the group A is torsion free, every element of A has a unique finite
presentation of the form
∑
i λiei with rational coefficients (although not
every such a rational combination belongs to A). Hence the subgroup H
of A given by the equation
∑
i λi = 0 is well defined, and the factor group
A/H is torsion free and has rank 1. Note that (M + H)/H ≃ M/(H ∩ M)
is infinite cyclic and A/(M + H) is the factor group of the torsion group
A/M . Therefore the group A/H is isomorphic to a subgroup of Dn.
It follows from the definition of H that for every i, the subgroup H
does not contain any non-zero multiple mgi of the element gi = em(i−1)+1.
Finally we set C = H ∩ K. Then A/C is embeddable in a finite direct
power of Dn since both A/H and A/K are. Assume now that C contains
a nontrivial normal in G subgroup L. Then L has to have infinite rank
since C ∩ R 6 K ∩ R = 0. Let T/L be the torsion part of A/L. Then
T is a normal in G subgroup with torsion free factor group A/T . Hence
T = Ni for some i. Therefore L and C must contain a non-zero multiple
of the element gi ∈ Ni. But H does not contain such elements, and the
statement (b) is proved by contradiction.
(c) The subgroup A is a direct sum of its Sylow p-subgroups Ap.
For every prime p, the elements x of Ap with px = 0 form a normal in
G subgroup A(p). It follows from the maximum condition for normal
subgroups of G that there are only finitely many primes p with nonzero
A(p). For the same reason, A has a finite exponent n. Arguing as in
the proof of (b), but replacing A by A(p) and taking the maximal set
(a1, a2, . . . ) in A(p) as there (but linearly independent over Z/pZ), we
obtain a subgroup (and subspace) Cp of finite codimension in A(p), which
does not contain any nonzero normal in G subgroup.
Now consider a maximal in Ap subgroup Ep such that Ep ∩A(p) = Cp.
Then every element x + Ep of order p from Ap/Ep must belong to the
A. Yu. Olshanski i 83
canonical image (A(p) + Ep)/Ep of the subgroup A(p) in Ap/Ep since
otherwise (〈x + Ep〉/Ep) ∩ ((A(p) + Ep)/Ep) is trivial, and so
〈x+Ep〉∩A(p) 6 Ep, contrary to the maximality of Ep. Since the subgroup
(A(p) + Ep)/Ep ≃ A(p)/Cp is finite, we have finitely many elements of
order p in the p-group Ap/Ep of finite exponent dividing n. Hence Ap/Ep
is a finite p-group.
If N is a non-trivial p-subgroup normal in G and N 6 Ep, then
N ∩ A(p) 6 Ep ∩ A(p) = Cp, where N ∩ A(p) is non-trivial and normal in
G, contrary to the choice of Cp. Thus Ep contains no such subgroups N .
Since Ap is a direct summand of A, for every Ep, one can find a
subgroup Fp 6 A with Ap ∩ Fp = Ep and A/Fp ≃ Ap/Ep. If a normal
in G subgroup N with nontrivial p-torsion were contained in Fp, then
Ap ∩ N 6 Ap ∩ Fp = Ep, which would provide a contradiction.
Now the intersection C = ∩p|nFp contains no nonzero subgroups of A,
which are normal in G. Since every A/Fp is a finite group of exponent
dividing n, the group A/C is embeddable in a finite direct power of the
group Z/nZ, as desired.
(a) Let T be the torsion subgroup of A. By the statement (b) applied
to G/T , we have a subgroup C ′ containing T but containing no bigger
normal in G subgroups, and A/C ′ is embeddable in a finite direct power
of some Dn. Note that C ′ contains no nontrivial torsion free subgroup N
normal in G since N + T > T .
Since T has a finite exponent m, it has a torsion free direct compliment
K in A by Kulikov’s theorem [11,14]. The intersection S of the subgroups
conjugated to K in G is torsion free, normal in G, and the exponent of
A/S is equal to the exponent of A/K ≃ T , i.e. it is m.
The statement (c) applied to G/S provides us with a subgroup C ′′
containing S but containing no bigger normal in G subgroups, with A/C ′′
embeddable in a finite direct power of Z/mZ. Note that C ′′ contains no
nontrivial torsion subgroup N normal in G since N + S > S.
It follows that C = C ′ ∩ C ′′ contains no nontrivial normal in G
subgroups and A/C is embeddable in a finite direct power of Dn ×Z/mZ.
Since both m and n can be replaced by their common multiple, the lemma
is proved.
Remark 3.3. One cannot generalize Lemma 3.2 to the slightly larger
class of central-by-metabelian groups. Indeed, every countable abelian
group A embeds as a central subgroup in some finitely generated central-
by-metabelian group G [3], and so every subgroup C of A becomes normal
in G.
84 On Kaluzhnin-Krasner’s embedding of groups
Proof of Theorem 3.1. (b) By Lemma 3.2 (b) and Proposition 1.1,
for some n, m > 1, the group G embeds in a wreath product W = D Wr B,
where D is a direct product of m copies of the group Dn. Since B is
infinite, it follows from the Fundamental Theorem of finitely generated
abelian groups, that B contains a subgroup of index m isomorphic to B.
In other words, B is a subgroup of index m in a group B0 isomorphic to
B. The group W0 = Dn Wr B0 = B0 ⋉ F , where F is the subgroup of
functions B0 → Dn, contains the subgroup W1 = B ⋉ F , and it remains
to show that W1 is isomorphic with W . This isomorphism is identical
on B and maps every function f0 : B0 → Dn to the function f : B → D
given by the rule f(b) = (f(t1b), . . . , f(tmb)) ∈ D, where {t1, . . . , tm} is a
transversal to the subgroup B in B0.
(c,a) One should argue as in the proof of (b) but with reference to
the items (c) and (a) of Lemma 3.2 and with the group Dn replaced by
Z/nZ and Dn × Z/nZ, respectively.
4. Subgroups of (ZZZ/pZZZ) W r ZZZ
Let us fix a prime number p. For every prime q 6= p, there is a finite-
dimensional faithful, irreducible presentation of Z/qZ over the Galois field
Fp. Let Vq be the corresponding representation module. Each of these
representations lifts to a representation of an infinite cyclic group 〈b〉, so
the direct sum V = ⊕q 6=pVq is a Fp〈b〉-module. The action of 〈b〉 defines
the semidirect product G = 〈b〉 ⋉ V .
Lemma 4.1.
(1) Every finitely generated subgroup of G is finite-by-cyclic, i.e. G is
locally finite-by-cyclic.
(2) G contains 2ℵ0 non-isomorphic subgroups.
(3) There is a subgroup C in V such that V/C has order p and C
contains no nontrivial normal in G subgroups.
Proof. (1) It is easy to see that any finite subset of G is contained in a
subgroup 〈b〉 ⋉ U , where U is the direct sum of the subgroups Vqi
over
a finite subset of prime numbers qi. Since U is finite, the statement (1)
follows.
(2) Denote by HS the subgroup 〈b〉 ⋉ VS , where S is a set of prime
numbers q 6= p and VS = ⊕q∈SVq. Since different Vq-s are irreducible and
non-isomorphic Fp〈b〉-modules, every normal in HS finite subgroup of VS
A. Yu. Olshanski i 85
is VT = ⊕q∈T Vq for a finite subset T ⊂ S. The centralizer of VT in HS has
index
∏
q∈T q. It follows that the groups HS1
and HS2
are not isomorphic
for S1 6= S2, which implies the statement (2).
(3) Let form an Fp-basis (e1, e2, . . . ) as the union of the bases of the
subspaces Vq and define the subspace C by the equation
∑
i xi = 0 in the
coordinates. Then clearly V/C has order p and C contains none of the
summands Vq. Since every normal in G subgroup of V is a submodule of
the direct sum of some non-isomorphic irreducible Fp〈b〉-modules Vq, it
coincides with some VS , and the statement is proved.
The next theorem demonstrates that the structure of subgroups of
wreath products of cyclic groups is quite rich.
Theorem 4.2. The wreath product Zp WrZ contains 2ℵ0 non-isomorphic
subgroups H, where each H is locally finite-by-cyclic.
Proof. The group G from Lemma 4.1 is embeddable in Zp WrZ by the
property (3) of Lemma 4.1 and Proposition 1.1. Therefore Theorem 4.2
follows from the properties (1) and (2) of Lemma 4.1.
Remark 4.3. Similar approach shows that the wreath product ZWrZ
contains 2ℵ0 countable locally polycyclic, non-isomorphic subgroups. But
now, instead of Vq, one should start with Z〈b〉-modules Vi, which are
non-isomorphic, finite-dimensional and irreducible over Q.
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Contact information
Alexander Yu.
Olshanskii
Department of Mathematics, Vanderbilt University,
Nashville, TN 37240, USA, and
Department of Mathematics and Mechanics,
Moscow State University, Moscow 119991, Russia
E-Mail(s): alexander.olshanskiy@vanderbilt.edu
Received by the editors: 03.02.2015
and in final form 25.02.2015.
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