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 , Theorem E.3). If Fπ ⊆ X ⊆ LFπ, then (HX)Cis 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 . Anti-ˇ CC-groups and anti-(PF)C-groups were described in . This line of research goes back to  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 hasX-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∈IAiis 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 andA = 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 hasX-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.
 V. S. Charin, D. I. Zaitsev, Groups with finiteness conditions and other restric- tions for subgroups, Ukrainian Math. J., 40, 1988, pp.233–241.
 S. N. Chernikov,Groups with given properties of a system of subgroups, Modern  B. Hartley, A dual approach to Chernikov modules, Math. Proc. Cambridge Phil.
 S. Franciosi, F. de Giovanni, M. J. Tomkinson, Groups with polycyclic-by-finite conjugacy classes, Boll. U.M.I., 4B, 1990, pp.35–55.
 S. Franciosi, F. de Giovanni, L. A. Kurdachenko, On groups with many almost normal subgroups, Ann. Mat. Pura Appl., CLXIX, 1995, pp.35–65.
 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.
 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 classes of subgroups, Comm. Algebra, 32, 2004, pp.4769–4784.
al, Groups with Chernikov classes of conjugate sub- groups, J. Group Theory, 54, 2005, pp.93–108.
 L. A. Kurdachenko, J. M. Munoz Escolano, J. Ot´  R. Maier, J. C. Rog´erio XC-elements in groups and Dietzmann classes, Beitr¨ Algebra Geom., 40, 1999, pp.243–260.
 R. Maier, Analogues of Dietzmann’s Lemma, In: Advances in Group Theory, Naples (Italy), Aracne Ed., 2002, pp.43–69.
 R. Maier, The Dietzmann property of some classes of groups with locally finite conjugacy classes, J. Algebra, 277, 2004, pp.364–369  G. A. Miller, H. C. Moreno Non-abelian groups in which every subgroup is abelian, Trans. Amer. Math. Soc., 4, 1903, pp.398–404.
 Ya. D. Polovickii, The groups with extremal classes of conjugate elements, Siberian Math. J., 5, 1964, Journal Algebra Discrete Math.
 D. J. Robinson. Finiteness conditions and generalized soluble groups, Vol. I and  F.G. Russo, Groups with soluble minimax conjugate classes of subgroups, Mashhad Research J. Math. Sci., 1, 2007, pp.41–49.
 F.G. Russo, Anti-CC-Groups and Anti-P C-Groups, Int. J. Math. Math. Sciences,  R. Schmidt, Subgroup lattices of groups, de Gruyter, Berlin, 1994.
 D. I. Zaitsev, On locally soluble groups with finite rank, Doklady A. N. SSSR,  D. I. Zaitsev, On the properties of groups inherited by their normal subgroups, Ukrainian Math. J., 38, 1986, pp.707–713.
Department of Mathematics, University ofNaples Federico II, via Cinthia I-80126,Naples, ItalyE-Mail: francesco.russo@dma.unina.itURL: russodipmatunina.altervista.org Received by the editors: 25.02.2010and in final form 25.02.2010.