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## Adm-n1.dvi

c Journal “Algebra and Discrete Mathematics”
A generalization of groups with many almost

*Dedicated to Professor I.Ya. Subbotin*
*on the occasion of his 60-th birthday*
A subgroup H of a group G is called almost
normal in G if it has finitely many conjugates in G. A classicresult of B. H. Neumann informs us that |G : Z(G)| is finite ifand only if each H is almost normal in G. Starting from thisresult, we investigate the structure of a group in which each non-finitely generated subgroup satisfies a property, which is weaker tobe almost normal.

In this paper X denotes an arbitrary class of groups which is closed withrespect to forming subgroups and quotients, F is the class of all finitegroups, Fπ is the class of all finite π-groups (π set of primes), ˇ
class of all Chernikov groups, PF is the class of all polycyclic-by-finitegroups, S2F is the class of all (soluble minimax)-by-finite groups. Givena positive integer r, we recall that the operator L, defined by
(1.1) LX = {G | g1, g2, . . . , gr ∈ X, ∀g1, g2, . . . gr ∈ G},
from X to X is called local operator for X. See [12, §C, p.54]. We recallthat the o Journal Algebra Discrete Math.

perator H, which associates to X the class of hyper-X-groups
This paper is dedicated to the memory of my father and to the future of my brother.

2010 Mathematics Subject Classification: 20C07; 20D10; 20F24.

Key words and phrases: Dietzmann classes; anti-XC-groups; groups with X-
classes of conjugate subgroups; Chernikov groups.

A generalization of groups with many .

is called extension operator. See [12, §E, p.60]. The notation follows[11, 12, 13, 16].

A subgroup H of a group G is called almost normal in G if H has
finitely many conjugates in G, that is, if |G : NG(H)| is finite. Neumann’sTheorem [16, Chapter 4, Vol.I, p.127] shows that G has each H which isalmost normal in G if and only if G/Z(G) ∈ F. We have NG(ClG(H)) =coreG(NG(H)) =
of conjugates of H in G. |G : NG(H)| = |ClG(H)| is finite if and only ifG/coreG(NG(H)) ∈ F. In [8, 9] G has F-classes of conjugate subgroups,if G/coreG(NG(H)) ∈ F for each H in G. Thus Neumann’s Theorem canbe reformulated, stating that G has G/coreG(NG(H)) ∈ F for each Hin G if and only if G/Z(G) ∈ F. See [9, Introduction]. More generally,G has X-classes of conjugate subgroups, if G/coreG(NG(H)) ∈ X foreach H in G. [9, Main Theorem] describes groups having ˇ
conjugate subgroups. [8, Main Theorem] describes those having PF-classes of conjugate subgroups.

Recall that ZX(G) = {x ∈ G | G/CG( x G) ∈ X} is a characteristic
subgroup of G, called XC-center of G. See [12, Definition B.1, Proposi-tion B.2]. G is called XC-group if it coincides with its XC-center. FC-groups, ˇ
CC-groups, (PF)C-groups and (S2F)C-groups are well–known
and described in [4, 7, 11, 12, 13, 15].

If G has F-classes of conjugate subgroups, then it is an FC-group.

C-classes of conjugate subgroups, then it
CC-group. From [8, Corollary 2.7], if G has PF-classes of conjugate
subgroups, then it is a (PF)C-group. From [17, Lemma 2.4], if G hasS2F-classes of conjugate subgroups, then it is an (S2F)C-group. Thenext lemma allows us to generalize these facts.

Lemma 1.1.

*Assume that *FX = X

*. If *G

*has *X

*-classes of conjugatesubgroups, then *ZX(G) = G

*.*
*Proof. *Let g ∈ G. G/H ∈ X, where H = coreG(NG( g )). Let H1 =CH( g ) and H2 = coreG(H1) = CH( g G). It is enough to proveG/H2 ∈ X. Of course, H ≥ NH( g ). Conversely, an element of NH( g )is an element of G, fixing g x = gx by conjugation for every x ∈ G,again fixing g by conjugation. If x = 1, then we get the elements ofH and so H ≤ N Journal Algebra Discrete Math.

H ( g ). Then H/H1 = NH ( g )/CH ( g ) is isomor-
phic to a subgroup of the automorphism group of g and so it is finite.

The same is true if we consider H1/H2 and NG( g )/CG( g ). Therefore,G/H2 is an extension of the finite group H1/H2 by the finite group H/H1by G/H ∈ X. From (FF)X = FX = X, G/H2 ∈ X.

We recall that X is called Dietzmann class, if for every group G and
x ∈ G, the following implication is true:
(1.2) if x ∈ ZX(G) and x ∈ X, then x G ∈ X,
See [12, Definitions B.1 and B.6]. Dietzmann classes are studied in [11,12, 13]. FC-groups form a Dietzmann class [12, Proposition D.3, b)]. Inparticular, this is true for periodic (PF)C-groups, which are obviouslyFC-groups. Note that F is a Dietzmann class [12, Proposition B.7, b)],but PF is not a Dietzmann class [12, Example B.8, c)]. Unfortunately, itis not known whether (PF)C-groups, ˇ
a Dietzmann class. See [4, 7, 11, 12, 13, 15]. But, they extend locally theclass of FC-groups. Therefore, the next result is significant.

Theorem 1.2 (see [12], Theorem E.3).

*If *Fπ ⊆ X ⊆ LFπ

*, then *(HX)C

*is a Dietzmann class.*
From Lemma 1.1, if X = F, then FC is a Dietzmann class. From
Lemma 1.1 and Theorem 1.2, if Fπ ⊆ X ⊆ LFπ, then (HX)C is a Diet-zmann class. Therefore, it is meaningful to ask whether we may weakenthe Neumann’s Theorem, looking at the following property for G:
(1.3) if H is non-finitely generated, then G/coreG(NG(H)) ∈ X, where
G is called anti-XC-group if it satisfies (1.3). Anti-FC-groups were de-scribed in [5]. Anti-ˇ
CC-groups and anti-(PF)C-groups were described
in [18]. This line of research goes back to [14] and deals with the struc-ture of groups with given properties of a system of subgroups.

We omit the elementary proofs of the next two results.

Lemma 2.1.

*Subgroups and quotients of anti-*XC

*-groups are anti-*XC

*-groups.*
Lemma 2.2.

*If *G

*is an anti-*XC

*-group and *ZX(G) = G

*, then *G

*has*X

*-classes of conjugate subgroups.*
Lemma 2Journal Algebra Discrete Math.

.3.

*Assume that *x

*is an element of the anti-*XC

*-group *G

*. If*
A = Dri∈IAi

*is a subgroup of *G

*consisting of *x

*-invariant nontrivialdirect factors *Ai

*, *i ∈ I

*, with infinite index set *I

*, then *x

*belongs to *ZX(G)

*.*
A generalization of groups with many .

*Proof. *This follows by [18, Lemma 3.3, Proof], considering X and ZX(G).

Corollary 2.4.

*Assume that *G

*is an anti-*XC

*-group and *A = Dri∈IAi

*is a subgroup of *G

*consisting of infinitely many nontrivial direct factors.*

Then A

*is contained in *ZX(G)

*.*
Lemma 2.5.

*Assume that *g

*is an element of the anti-*XC

*-group *G

*and*A = Dri∈IAi

*is a subgroup of *G

*, with *I

*as in Lemma 2.3. If *g ∈ NG(A)

*and *gn ∈ CG(A)

*for some positive integer *n

*, then *g

*belongs to *ZX(G)

*.*
*Proof. *This follows by [18, Lemma 3.7, Proof], considering X and ZX(G).

Corollary 2.6.

*If the anti-*XC

*-group *G

*has an abelian torsion subgroupthat does not satisfy the minimal condition on its subgroups, then allelements of finite order belong to *ZX(G)

*.*
*Proof. *This follows by [18, Corollary 3.9, Proof], considering X and ZX(G).

Theorem 2.7.

*If *G

*is a locally finite anti-*XC

*-group, then either *G

*has*X

*-classes of conjugate subgroups or *G

*is a Chernikov group.*
*Proof. *This follows by [18, Theorem 3.12, Proof], considering X andZX(G).

Note that Theorem 2.7 improves [18, Theorems 3.11 and 3.12].

Lemma 2.8.

*Assume that *X

*is residually closed. If *G

*has *X

*-classes ofconjugate subgroups, then *G ∈ N2X

*, where *N2

*is the class of nilpotentgroups of class at most 2.*
NG(H) be the norm of G. N(G) ≤ Z2(G) from
a result of Schenkman [19, Corollary 1.5.3]. Since G has X-classes ofconjugate subgroups, G/N(G) is residually X and so G/N(G) ∈ X. Thisgives as claimed.

Corollary 2.9.

*As *Journal Algebra Discrete Math.

*sume that *X

*is residually closed. If *G

*is a locally finite*
*anti-*XC

*-group, then either *G ∈ N2X

*or *G

*is a Chernikov group.*
*Proof. *This follows by Theorem 2.7 and Lemma 2.8.

Recall that G has f inite abelian section rank if it has no infinite elemen-tary abelian p-sections for every prime p (see [16, Chapter 10, vol.II]).

Following [5, 16, 20], a soluble-by-finite group G is an S1-group if ithas finite abelian section rank and the set of prime divisors of orders ofelements of G is finite.

Theorem 3.1.

*Assume that *X

*is residually closed. Let *G

*be an anti-*XC

*-group having an ascending series whose factors are either locally nilpotentor locally finite. Then either *G

*has *X

*-classes of conjugate subgroups or isa soluble-by-finite *S1

*-group or has a normal soluble *S1

*-subgroup *K

*suchthat *G/K ∈ X

*.*
*Proof. *G has an ascending normal series whose factors are either locallynilpotent or locally finite by [16, Theorem 2.31]. Let K be the largestradical normal subgroup of G. From Lemma 2.1 and Corollary 2.9, thelargest locally finite normal subgroup T /K of G/K is either a Chernikovgroup or in N2X.

In the first case, if H/T is a locally nilpotent normal subgroup of
G/T , then CH/K(T/K) is a locally nilpotent normal subgroup of G/K,so CH/K(T/K) is trivial and H/K is a Chernikov group. Then T = Gand so G has a normal radical subgroup K such that T /K is a Chernikovgroup (in this situation G is said to be a radical-by-Chernikov group).

In the second case, T /K = (N/K)(L/K), where N/K ∈ N2 is a
normal subgroup of T /K such that (T /K)/(N/K) ∈ X. If N/K isnontrivial, then there exists a nontrivial element xK ∈ N/K such that
xK G = x GK/K is a nilpotent normal subgroup of G/K contained in
T /K. Since G/K has no nontrivial locally nilpotent normal subgroups,we get to a contradiction. Therefore N/K is trivial and T /K ∈ X. Thenwe may deduce as above that G has a normal radical subgroup K suchthat T /K ∈ X (in this situation G is said to be a radical-by-X group).

Assume that G has X-classes of conjugate subgroups. Then every
abelian subgroup of G has finite total rank by Corollary 2.4. A resultof Charin [16, Theorem 6.36] implies that K is a soluble S1-group. Weconclude that G has a normal soluble S1-subgroup K such that G/K isa Chernikov group. Therefore G is an extension of a soluble S1-group byan abelian group with min by a finite group. An abelian group with minis clearly an S1-group and the class of S1-groups is closed with respectto extensioJournal Algebra Discrete Math.

ns of two of its members (see [16, Chapter 10]). Therefore G is
a soluble-by-finite S1-group. The remaining case is that G has a normalsoluble S1-subgroup K such that G/K ∈ X.

Note that Theorem 3.1 improves [18, Theorems 4.1 and 4.2].

A generalization of groups with many .

Corollary 3.2.

*Assume that *X

*is residually closed. Let *G

*be an anti-*XC

*-group having an ascending series whose factors are either locally nilpotentor locally finite. Then either *G ∈ N2X

*or *G

*is a soluble-by-finite *S1

*-groupor *G

*has a normal soluble *S1

*-subgroup *K

*such that *G/K ∈ X

*.*
*Proof. *This follows by Theorem 3.1 and Corollary 2.9.

[1] V. S. Charin, D. I. Zaitsev, Groups with finiteness conditions and other restric-
tions for subgroups, Ukrainian Math. J., 40, 1988, pp.233–241.

[2] S. N. Chernikov,Groups with given properties of a system of subgroups, Modern
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conjugacy classes, Boll. U.M.I., 4B, 1990, pp.35–55.

[5] S. Franciosi, F. de Giovanni, L. A. Kurdachenko, On groups with many almost
normal subgroups, Ann. Mat. Pura Appl., CLXIX, 1995, pp.35–65.

[6] H. Heineken, L. A. Kurdachenko, Groups with Subnormality for All Subgroups
that Are Not Finitely Generated, Ann. Mat. Pura Appl., CLXIX, 1995, pp.203–232.

[7] L. A. Kurdachenko, On groups with minimax conjugacy classes, In: Inﬁnite
groups and adjoining algebraic structures, Kiev (Ukraine), Naukova Dumka, 1993,pp.160–177.

al, P. Soules, Groups with polycyclic-by-finite conjugate
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al, Groups with Chernikov classes of conjugate sub-
groups, J. Group Theory, 54, 2005, pp.93–108.

[10] L. A. Kurdachenko, J. M. Munoz Escolano, J. Ot´
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[12] R. Maier, Analogues of Dietzmann’s Lemma, In: Advances in Group Theory,
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[13] R. Maier, The Dietzmann property of some classes of groups with locally finite
conjugacy classes, J. Algebra, 277, 2004, pp.364–369
[14] G. A. Miller, H. C. Moreno Non-abelian groups in which every subgroup is abelian,
Trans. Amer. Math. Soc., 4, 1903, pp.398–404.

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Math. J., 5, 1964, Journal Algebra Discrete Math.

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[17] F.G. Russo, Groups with soluble minimax conjugate classes of subgroups, Mashhad
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[18] F.G. Russo, Anti-CC-Groups and Anti-P C-Groups, Int. J. Math. Math. Sciences,
[19] R. Schmidt, Subgroup lattices of groups, de Gruyter, Berlin, 1994.

[20] D. I. Zaitsev, On locally soluble groups with finite rank, Doklady A. N. SSSR,
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Department of Mathematics, University ofNaples Federico II, via Cinthia I-80126,Naples, Italy

*E-Mail: *francesco.russo@dma.unina.it

*URL: *russodipmatunina.altervista.org
Received by the editors: 25.02.2010and in final form 25.02.2010.

Source: http://adm.lnpu.edu.ua/downloads/issues/2010/N1/adm-n1-6.pdf

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