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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 14 (2012). Number 1. pp. 37 – 48
c© Journal “Algebra and Discrete Mathematics”
On locally soluble AFN-groups
Olga Yu. Dashkova
Communicated by L. A. Kurdachenko
Abstract. Let A be an RG-module, where R is a com-
mutative ring, G is a locally soluble group, CG(A) = 1, and each
proper subgroup H of G for which A/CA(H) is not a noetherian
R-module, is finitely generated. We describe the structure of a
locally soluble group G with these conditions and the structure of
G under consideration if G is a finitely generated soluble group and
the quotient module A/CA(G) is not a noetherian R-module.
Introduction
Let A be a vector space over a field F , GL(F,A) be the group of
all automorphisms of A. Subgroups of GL(F,A) are called linear groups.
If A has a finite dimension over F , GL(F,A) can be considered as a
group of non-singular (n× n)-matrixes over F , where n = dimFA. Finite
dimensional linear groups have been studied by many authors. In the case
when A has infinite dimension over F , the situation is rather different.
Infinite dimensional linear groups were investigated a little. Study of this
class of groups requires some finiteness conditions. The one from these
finiteness conditions is a finitarity of infinite dimensional linear group.
We recall that a linear group is called finitary if for each element g ∈ G
the subspace CA(g) has finite codimension in A (see [1], [2], for example).
Many results have been obtained conserning finitary linear groups [2].
In [3] antifinitary linear groups are investigated. Let G ≤ GL(F,A),
A(wFG) be the augmentation ideal of the group ring FG, augdimF (G) =
2010 MSC: 20F19.
Key words and phrases: locally soluble group, noetherian module, group ring.
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38 On locally soluble AFN-groups
dimF (A(wFG)). A linear group G is called antifinitary if each proper
subgroup H of infinite dimension augdimF (H) is finitely generated [3].
If G ≤ GL(F,A) then A can be considered as an FG-module. The
natural generalization of this case is a consideration of an RG-module
A where R is a ring. B.A.F. Wehrfritz have considered artinian-finitary
groups of automorphisms of a module M over a ring R and noetherian-
finitary groups of automorphisms of a module M over a ring R which are
the analogues of finitary linear groups [4, 5, 6]. A group of automorphisms
F1AutRM of a module M over a ring R is called artinian-finitary if
A(g − 1) is an artinian R-module for each g ∈ F1AutRM . A group of au-
tomorphisms FAutRM of a module M over a ring R is called noetherian-
finitary if A(g − 1) is a noetherian R-module for each g ∈ FAutRM .
B.A.F. Wehrfritz have investigated the relation between F1AutRM and
FAutRM [6].
In [7] the notion of the cocentralizer of a subgroup H in the module
A have been introduced. Let A be an RG-module where R is a ring, G
is a group. If H ≤ G then A/CA(H) considered as an R-module is called
the cocentralizer of a subgroup H in A.
In this paper we consider the analogue of antifinitary linear groups
in theory of modules over group rings. Let A be an RG-module where
R is a ring, G is a group. We say that a group G is an AFN-group if
each proper subgroup H of G for which A/CA(H) is not a noetherian
R-module, is finitely generated.
In the paper locally soluble AFN-groups are investigated. Later on it is
considered RG-module A such that R is a commutative ring, CG(A) = 1.
The main results ate theorems 1, 2. In theorem 1 the structure of a
locally soluble AFN-group is described. In theorem 2 the structure of a
finitely generated soluble AFN-group G is described in the case where
the cocentralizer of G in A is not a noetherian R-module.
1. Prelimlnary results
We begin by assembling some elementary facts about AFN-groups.
Lemma 1. Let A be an RG-module.
(1) If L ≤ H ≤ G and the cocentralizer of a subgroup H in A is a
noetherian R-module, then the cocentralizer of a subgroup L in A is a
noetherian R-module.
(2) If L,H ≤ G and the cocentralizers of subgroups L, H in A are
noetherian R-modules, then the cocentralizer of 〈L,H〉 in A is a noetherian
R-module.
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O. Yu. Dashkova 39
Corollary 1. Let A be an RG-module, ND(G) be a set of all elements
x ∈ G such that the cocentralizer of 〈x〉 in A is a noetherian R-module.
Then ND(G) is a normal subgroup of G.
Proof. By lemma 1 ND(G) is a subgroup of G. Since CA(xg) = CA(x)g
for all x, g ∈ G then ND(G) is a normal subgroup of G.
Corollary 2. Let A be an RG-module, G be an AFN-group. If G has
proper non-finitely generated subgroups K and L then the cocentralizer of
〈K,L〉 in A is a noetherian R-module.
Lemma 2. Let A be an RG-module, G be an AFN-group. Suppose
that H is a subgroup of G and K is a normal subgroup of H such that
H/K = Drλ∈Λ(Hλ/K) where Hλ 6= K for every λ ∈ Λ and the index set
Λ is infinite. Then the cocentralizer of H in A is a noetherian R-module.
Proof. The quotient group H/K is decomposed in the direct product
H/K = H1/K × H2/K such that H1/K and H2/K are non-finitely
generated quotient groups. Since G is an AFN-group then by Lemma 1
the cocentralizer of H in A is a noetherian R-module.
Corollary 3. Let A be an RG-module, G be an AFN-group. Suppose
that H is a subgroup of G and K is a normal subgroup of H such that
H/K = Drλ∈Λ(Hλ/K), Hλ 6= K for every λ ∈ Λ and the index set Λ is
infinite. If g is an element of G such that Hλ is 〈g〉-invariant for every
λ ∈ Λ, then g ∈ ND(G).
Proof. The subgroup K is 〈g〉-invariant. Since the index set Λ is infinite,
Drλ∈Λ(Hλ/K)〈gK〉 = (H1/K)((H2/K)〈gK〉),
where H1 and H2〈g〉 are proper non-finitely generated subgroups of G.
It follows that the cocentralizer of 〈H, g〉 in A is a noetherian R-module.
By lemma 1 the cocentralizer of 〈g〉 in A is a noetherian R-module.
Corollary 4. Let A be an RG-module, G be an AFN-group. Suppose
that H is a subgroup of G and K is a normal subgroup of H such that
H/K = Drλ∈Λ(Hλ/K), Hλ 6= K for every λ ∈ Λ and the index set Λ is
infinite. If Hλ is G-invariant for every λ ∈ Λ, then G = ND(G).
Corollary 5. Let A be an RG-module, G be an AFN-group. Suppose
that H is a subgroup of G and K is a normal subgroup of H such that
H/K is an infinite elementary abelian p-group for some prime p. If g is
an element of G such that H and K are 〈g〉-invariant and gk ∈ CG(H/K)
for some k ∈ N then g ∈ ND(G).
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40 On locally soluble AFN-groups
Proof. Let 1 6= h1K ∈ H/K,H1/K = 〈h1K〉〈gK〉. Since the element g
induced on the quotient group H/K an automorphism of finite order,
H1/K is finite. Since the quotient group H/K is elementary abelian then
H/K = H1/K × C1/K. Note that the set {Cy1 |y ∈ 〈g〉} is finite. Let
{Cy1 |y ∈ 〈g〉} = {U1, · · · , Um}.
Then the 〈g〉-invariant subgroup D1 = U1 ∩ · · · ∩ Um = Core〈g〉(C1)
has finite index in H. Moreover, since the subgroup K is 〈g〉-invariant,
K ≤ D1. Let 1 6= h2K ∈ D1/K,H2/K = 〈h2K〉〈gK〉. Then
〈H1/K,H2/K〉 = H1/K ×H2/K.
Again we have H/K = (H1/K ×H2/K)× C2/K for some subgroup C2.
Reasoning in a similar way, we construct an infinite family {Hn/K|n ∈ N}
of non-identity 〈g〉-invariant subgroups such that
〈Hn/K|n ∈ N〉 = Drn∈NHn/K.
By corollary 3 g ∈ ND(G).
2. On locally soluble AFN-groups
A group G is said to have finite 0-rank r0(G) = r if G has a finite
subnormal serires with exactly r infinite cyclic factors, all other factors
being periodic. It is well known that the 0-rank is independent of the
chosen series.
Lemma 3. Let A be an RG-module, G be an AFN-group. Suppose that
a group G has a normal subgroup K such that G/K is an abelian quotient
group of infinite 0-rank. Then the cocentralizer of G in A is a noetherian
R-module.
Proof. Let B/K be a free abelian subgroup of G/K such that G/B is
periodic. If π(G/B) is infinite then the cocentralizer of G in A is a
noetherian R-module by lemma 2. Suppose that π(G/B) is finite and
choose a prime q such that q 6∈ π(G/B). Put C/K = (B/K)q so that
B/C is a Sylow q-subgroup of G/C. Let P/C be the Sylow q′-subgroup
of G/C. Then G/P is an infinite elementary abelian q-group. By lemma 2
the cocentralizer of G in A is a noetherian R-module.
Corollary 6. Let A be an RG-module, G be an AFN-group. Suppose
that G has a normal subgroup K such that G/K is an abelian-by-finite
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O. Yu. Dashkova 41
group of infinite 0-rank. Then the cocentralizer of G in A is a noetherian
R-module.
Proof. Let L/K be a normal abelian subgroup of G/K such that G/L
is finite. Then r0(L/K) is infinite. Pick g ∈ G\L. Let B/K be a free
abelian subgroup of L/K such that the quotient group L/B is periodic.
The rank r0(B/K) is infinite. Choose an element a1 ∈ B\K. Put A1/K =
(〈a1〉K/K)〈gK〉. Since G/L is finite, A1/K is a finitely generated abelian
group. It follows that A1/K ∩ B/K is finitely generated. Choose the
subgroup C1/K of B/K which maximal under
(A1/K ∩B/K) ∩ C1/K = 〈1〉.
Then L/C1 is a group of finite 0-rank. Since G/L is finite, the family
{(C1/K)yK |y ∈ 〈g〉} is finite. Let
{(C1/K)yK |y ∈ 〈g〉} = {D1/K, · · · , Dn/K},
and put
E/K = D1/K ∩ · · · ∩ Dn/K.
Then E/K ≤ B/K, E/K is 〈g〉-invariant. By Remak’s theorem L/E
has finite 0-rank. In particular, E/K has infinite 0-rank. Choose an
element a2 ∈ E\K. Put A2/K = (〈a2〉K/K)〈gK〉. Then A2/K ≤ E/K,
(A1/K)∩ (A2/K) = 1. Proceeding in the same way, we construct a family
{An/K|n ∈ N} of non-identity 〈g〉-invariant subgroups such that
〈An/K|n ∈ N〉 = Drn∈N(An/K).
By corollary 3 g ∈ ND(G). We can choose a finitely generated subgroup
F of G such that G/K = (FK/K)(L/K) and for each element g of F
g ∈ ND(G). Since F is a finitely generated subgroup then F ≤ ND(G).
By lemma 3 the cocentralizer of L in A is a noetherian R-module. Since
G = FL then by lemma 1 the cocentralizer of G in A is a noetherian
R-module.
Lemma 4. Let A be an RG-module, G be an AFN-group. Suppose that
G has subgroups L ≤ K ≤ H such that L and K are normal subgroups of
H, K/L is a divisible Chernikov group and H/K is a polycyclic-by-finite
group. If the cocentralizer of H in A is not a noetherian R-module, then
H = G. Moreover, either G = K (so that G/L is a Pru¨fer p-group for
some prime p) or G/K is a cyclic q-group for some prime q.
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42 On locally soluble AFN-groups
Proof. Suppose that H/L is finitely generated. By P. Hall theorem (theo-
rem 5.34 [8]) H/L satisfies the maximal condition for normal subgroups.
In particular, K/L satisfies the condition max−H. Since K/L is a di-
visible Chernikov group, this is impossible. Therefore H/L c n not be
finitely generated and thus H is non finitely generated subgroup. Since
the cocentralizer of H in A is not a noetherian R-module, then H = G.
Suppose that G 6= K. Then G = 〈K,M〉 for some finite set M . Since
M is finite, we may choose a subset S of M such that G = 〈K,S〉 but
G 6= 〈K,X〉 for any proper subset X of S. Let
S = {x1, · · · , xm}.
If m > 1, then 〈K,x1, · · · , xm−1〉 and 〈K,xm〉 are proper non finitely gen-
erated subgroups of G. Since G is an AFN-group then the cocentralizers of
subgroups 〈K,x1, · · · , xm−1〉 and 〈K,xm〉 in A are noetherian R-modules.
Since G = 〈〈K,x1, · · · , xm−1〉, 〈K,xm〉〉, by lemma 1 the cocentralizer of
G in A is a noetherian R-module. This is a contradiction that shows that
m = 1. Therefore G/K = 〈xK〉 is cyclic. If G/K is infinite, then G must
be a product of two proper non finitely generated subgroups, what again
gives a contradiction. If G/K is finite but |π(G/K)| > 1 , we again have
a contradiction. Hence G/K is a cyclic q-group for some prime q.
Lemma 5. Let A be an RG-module, G be an AFN-group. Suppose that
H is a normal subgroup of G such that G/H is an infinite abelian-by-
finite periodic group. If the cocentralizer of G in A is not a noetherian
R-module, then either G/H is a Pru¨fer p-group for some prime p or G
has a normal subgroup K such that G/K is a cyclic q-group for some
prime q, H ≤ K and K/H is a Chernikov divisible p-group for some
prime p.
Proof. Let L/H be an abelian normal subgroup of G/H such that G/L is
finite. If π(L/H) is infinite, then the cocentralizer of L in A is a noetherian
R-module by lemma 2. By corollary 4 G = ND(G). Since G/L is finite,
it follows that the cocentralizer of G in A is a noetherian R-module by
lemma 1. This contradiction proves that π(L/H) is finite. Then there
exists a prime p such that the Sylow p-subgroup P/H of L/H is infinite.
Let F/H be the Sylow p′-subgroup of L/H. There is a finite subgroup S/H
such that G/H = (L/H)(S/H). If F/H is infinite then both subgroups
(P/H)(S/H) and (F/H)(S/H) are not finitely generated. Therefore the
cocentralizers of subgroups PS and FS in A are noetherian R-modules.
By lemma 1 the cocentralizer of G in A is a noetherian R-module. This
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O. Yu. Dashkova 43
contradiction shows that F/H is finite. Put B/H = (P/H)p. If P/B is
infinite then P/B is not finitely generated. Therefore the cocentralizer
of P in A is a noetherian R-module. By corollary 5 G = ND(G). Since
G/P is finite, it follows that the cocentralizer of G in A is a noetherian
R-module by lemma 1. This contradiction proves that (P/H)/(B/H) is
finite. By lemma 3 [9] P/H = (V/H) × (D/H) where D/H is divisible
and V/H is finite. D is a G-invariant subgroup. Put K = D. Since G/D
is finite, it is suffices to apply lemma 4.
Lemma 6. Let A be an RG-module, G be an AFN-group. Suppose that
G has normal subgroups K ≤ H such that G/H is finite and H/K is
torsion-free abelian. If the cocentralizers of G in A is not a noetherian
R-module, then H/K is finitely generated.
Proof. By corollary 6 H/K has finite 0-rank. Let B/K be a free abelian
subgroup of H/K such that H/B is periodic. Since r0(H/K) is finite then
B/K is finitely generated. Suppose that H/K is not finitely generated.
Since G/H is finite, C/K = (B/K)G/K is finitely generated. By lemma 5
|π(G/C)| ≤ 2. Choose the distinct primes r, s such that r, s 6∈ π(G/C).
Put D/K = (C/K)rs. Then G/D is abelian-by-finite, periodic and not
finitely generated. Moreover |π(G/D)| ≥ 3. This contradicts lemma 5.
Therefore H/K is finitely generated.
Lemma 7. Let A be an RG-module, G be an AFN-group. Suppose that
G has two normal subgroups K ≤ H such that G/H is finite and H/K
is abelian and not finitely generated. If the cocentralizer of G in A is not
a noetherian R-module, then H/K is Chernikov.
Proof. By corollary 6 H/K has finite 0-rank. Let T/K be the periodic part
of H/K. By lemma 6 H/T is finitely generated. Then H/K has a finitely
generated subgroup B/K such that H/B is periodic. Since G/H is finite,
C/K = (B/K)G/K is finitely generated. By lemma 5 G/C is a Chernikov
group. It follows that T/K is Chernikov too. Let D/K be the divisible
part of T/K. Then G/D is finitely generated and abelian-by-finite. It is
suffices to apply lemma 4.
Lemma 8. Let A be an RG-module, G be a soluble AFN-group. If G is
not a Pru¨fer p-group for some prime p then G/ND(G) is a polycyclic
quotient group.
Proof. Put D = ND(G). If the cocentralizer of G in A is a noetherian
R-module, then G = ND(G). Therefore we suppose that G 6= ND(G).
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44 On locally soluble AFN-groups
Let D = D0 ≤ D1 ≤ · · · ≤ Dn = G be a series of subnormal
subgroups of G whose factors are abelian. Consider the factor Dj/Dj−1,
j < n. If this factor is not finitely generated, then the subgroup Dj cannot
be finitely generated and the cocentralizer of Dj in A is a noetherian
R-module. In particular,Dj ≤ ND(G). It follows that Dj/Dj−1 is finitely
generated for every j = 1, · · · , n− 1. Put K = Dn−1. If G/K is finitely
generated, then G/D is polycyclic, and all is done. Suppose that G/K
is not finitely generated. By lemma 7 G/K is a Chernikov group. Let
P/K be the divisible part of G/K. If P/K 6= G/K, then P is not finitely
generated proper subgroup of G. Thus the cocentralizer of P in A is a
noetherian R-module. Therefore P ≤ ND(G). But in this case G/ND(G)
is finite. Contradiction. Hence G/K = P/K. Clearly in this case G/K
is a Pru¨fer p-group for some prime p. Let g ∈ G\K. Since g 6∈ ND(G),
〈g,K〉 is finitely generated. The finiteness of 〈g〉K/K implies that K is
finitely generated (theorem 1.41 [8]). Since G is not a Pru¨fer p-group for
some prime p, then K 6= 1. It follows that K has a proper G-invariant
subgroup L of finite index such that G/L is Chernikov and not divisible.
As above, in this case G/ND(G) is finite.
Lemma 9. Let A be an RG-module, G be a locally soluble AFN-group.
If the cocentralizer of G in A is a noetherian R-module, then G contains
a normal hyperabelian subgroup N such that G/N is soluble.
Proof. Since the cocentralizer of G in A is a noetherian R-module, then
A/CA(G) is a finitely generated R-module. Put C = CA(G). A has the
finite series of RG-submodules
〈0〉 = C0 ≤ C1 = C ≤ C2 = A,
such that C2/C1 is a finitely generated R-module.
By theorem 13.5 [10] the quotient group G = G/CG(C2/C1) contains
a normal hyperabelian, locally nilpotent subgroup N = N/CG(C2/C1)
such that G/N is imbedded in the Cartesian product Πα∈AGα of finite
dimensional linear groups Gα of degree f ≤ n where n depends on the
number of generating elements of R-module C2/C1 only. Since G is a
locally soluble group then G is locally soluble too. It follows that the
projection Hα of G/N on each subgroup Gα is a locally soluble finite
dimensional linear group of degree at most n. By corollary 3.8 [10] Hα
is a soluble group for each α ∈ A. By theorem 3.6 [10] each group Hα
contains a normal subgroup Kα such that |Hα : Kα| ≤ µ(n), Kα is a
triangularizable group, Kα has a nilpotent subgroup Mα of step at most
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O. Yu. Dashkova 45
n− 1, Mα is a normal subgroup of Hα and Kα/Mα is abelian. Therefore
H = Πα∈AHα contains a normal nilpotent subgroup M = Πα∈AMα of
step at most n − 1, H/M has a normal abelian subgroup K/M where
K = Πα∈AKα and (H/M)/(K/M) is a locally finite group of the finite
period at most µ(n)!. It follows that H is a soluble group of the derived
length at most n− 1 + 1+ µ(n)! = n+ µ(n)!. Therefore G/N is a soluble
group of the derived length at most n+ µ(n)!. It follows that G has the
series of normal subgroups CG(C2/C1) ≤ N ≤ G. As G/N ≃ G/N then
G/N is a soluble group of the derived length at most n + µ(n)!. Since
CG(A/CA(G)) is abelian and N/CG(C2/C1) is hyperabelian then N is
hyperabelian too.
Theorem 1. Let A be an RG-module, G be a locally soluble AFN-group.
Then G has an ascending series of normal subgroups
〈1〉 = L0 ≤ L1 ≤ L2 ≤ · · · ≤ Lγ ≤ · · · ≤ Lδ = G
such that each factor Lγ+1/Lγ , γ < δ, is hyperabelian.
Proof. If the cocentralizer of G in A is a noetherian R-module then we
apply lemma 9. Later on we consider the case where the cocentralizer
of G in A is not a noetherian R-module. If G is a soluble group then
the theorem is valid. Let G be non soluble. By corollary 5.27 [8] G
cannot be simple. Therefore G has a proper normal subgroup H1. If H1
is finitely genetated, then it is soluble. It follows that H1 has the series of
G-admissible subgroups
〈1〉 = B0 ≤ B1 ≤ B2 ≤ · · · ≤ Bk = H1
such that the factors Bt/Bt−1, t = 1, · · · , k, are abelian. If H1 is not
finitely generated, then the cocentralizer of H1 in A is a noetherian R-
module. By lemma 9 H1 contains a normal hyperabelian subgroup N1 such
that H1/N1 is soluble. Then H1 has the series of G-admissible subgroups
〈1〉 = R0 ≤ R1 ≤ R2 ≤ · · · ≤ Rm = H1
such that the factors Rt/Rt−1, t = 2, · · · ,m, are abelian, R1 is a hyper-
abelian subgroup. If G/H1 is a soluble group, then G has the series of
normal subgroups
H1 = G0 ≤ G1 ≤ G2 ≤ · · · ≤ Gr = G
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46 On locally soluble AFN-groups
such that the factors Gt/Gt−1, t = 1, · · · , r, are abelian. Therefore G has
an ascending series of normal subgroups
〈1〉 = L0 ≤ L1 ≤ L2 ≤ · · · ≤ Ln = G,
such that each factor Lt/Lt−1, t = 1, · · · , n, is hyperabelian. If G/H1 is
not a soluble group, then G/H1 has a proper normal subgroup H2/H1.
As above H2/H1 has the series of G-admissible subgroups
H1 = D0 ≤ D1 ≤ D2 ≤ · · · ≤ Dj = H2
such that each factor Dt/Dt−1, t = 1, · · · , j, is hyperabelian.
We proceed in this way. At step with the ordinal α we have that G/Hα
is a soluble quotient group. It follows that G has an ascending series of
normal subgroups
〈1〉 = L0 ≤ L1 ≤ L2 ≤ · · · ≤ Lγ ≤ · · · ≤ Lδ = G
such that each factor Lγ+1/Lγ , γ < δ, is hyperabelian.
Lemma 10. Let A be an RG-module, G be a finitely generated soluble
AFN-group. Then the cocentralizer of ND(G) in A is a noetherian R-
module.
Proof. Put D = ND(G) and let
〈1〉 = D0 ≤ D1 ≤ · · · ≤ Dn = D
be the derived series of D. If each factor Dj+1/Dj , j = 0, 1, · · · , n− 1,
is finitely generated, then D is polycyclic, and, in particular, D is finitely
generated. By lemma 1 the cocentralizer of D in A is a noetherian
R-module. Therefore, we suppose that some of the factors Dj+1/Dj ,
j = 0, 1, · · · , n − 1, is not finitely generated. Let t be a number such
that Dt/Dt−1 is not finitely generated but Dj+1/Dj is finitely generated
for every j ≥ t. It follows that D/Dt is polycyclic. Since G is a finitely
generated group then Dt is a proper non finitely generated subgroup of G.
Therefore the cocentralizer of Dt in A is a noetherian R-module. Since
D/Dt is polycyclic, D = KDt for some finitely generated subgroup K.
As K ≤ ND(G), we have that the cocentralizer of K in A is a noetherian
R-module. By lemma 1 the cocentralizer of ND(G) in A is a noetherian
R-module.
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O. Yu. Dashkova 47
Theorem 2. Let A be an RG-module, G be a finitely generated soluble
AFN-group. If the cocentralizer of G in A is not a noetherian R-module,
then the following conditions holds:
(1) the cocentralizer of ND(G) in A is a noetherian R-module;
(2) G has the series of normal subgroups B ≤ R ≤ W ≤ G such that
B is abelian, R/B is locally nilpotent, W/R is nilpotent and G/W is a
polycyclic group.
Proof. By lemma 10 the cocentralizer of ND(G) in A is noetherian R-
module. Let C = CA(ND(G)). Since A/C is a noetherian R-module, then
A has the finite series of RG-submodules 〈0〉 = C0 ≤ C1 = C ≤ C2 = A,
such that A/C is a finite generated R-module.
By theorem 13.5 [10] the quotient group S = G/CG(C2/C1) contains
the normal locally nilpotent subgroup D = N/CG(C2/C1) such that the
quotient group S/D is embedded in the Cartesian product Πα∈AGα of
finite dimensional linear groups Gα of degree f ≤ n where n depends on
the number of generating elements of an R-module C2/C1 only. Since the
group G is soluble then the quotient group S is soluble too. Therefore the
projection Hα of S on each subgroup Gα is a soluble finite dimensional
linear group of degree at most n. By theorem 3.6 [10] each group Hα
contains the normal subgroup Kα such that |Hα : Kα| ≤ µ(n), the
subgroup Kα is triangularizable, Kα contains the nilpotent subgroup
Mα of step at most n − 1 such that Mα is a normal subgroup of Gα
and the quotient group Kα/Mα is abelian. Therefore H = Πα∈AHα
contains the normal nilpotent subgroup M = Πα∈AMα of step at most
n− 1, the quotient group H/M has the normal abelian subgroup K/M
where K = Πα∈AKα and the quotient group (H/M)/(K/M) is a locally
finite group of the period at most µ(n)!. Since S/D is embedded in
the Cartesian product H = Πα∈AHα then S has the series of normal
subgroups D ≤ L ≤ F ≤ S such that D is locally nilpotent, L/D is
nilpotent, F/L is abelian and S/F is a locally finite group of the finite
period. Since G is a finitely generated group then S is finitely generated
too. Therefore the quotient group S/F is finite. It follows that S/L
is an almost abelian group. Since S/L is finitely generated then S/L
is a polycyclic group. Therefore S has the series of normal subgroups
D ≤ L ≤ S such th t D is locally nilpotent, L/D is nilpotent, S/L is a
polycyclic group.
Let B = CG(C1) ∩ CG(C2/C1). Each element of B acts trivially in
each factor Cj+1/Cj , j = 0, 1. It follows that B is abelian. By Remak’s
theorem
G/B ≤ G/CG(C1)×G/CG(C2/C1).
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48 On locally soluble AFN-groups
As ND(G) ≤ CG(C1) then the quotient group G/CG(C1) is polycyclic
by lemma 8. Since S = G/CG(C2/C1) has the series of normal subgroups
D ≤ L ≤ S such that D is locally nilpotent, L/D is nilpotent, S/L is a
polycyclic group then G has the series of normal subgroups
B ≤ R ≤ W ≤ G
such that B is abelian, R/B is locally nilpotent, W/R is nilpotent and
G/W is a polycyclic group.
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Contact information
O. Yu. Dashkova 49055, Ukraine, Dnepropetrovsk, prospekt
Kirova, 102-D, kv.35.
E-Mail: odashkova@yandex.ru
Received by the editors: 21.04.2012
and in final form 02.10.2012.