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)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 [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 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.
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Department of Mathematics, University ofNaples Federico II, via Cinthia I-80126,Naples, ItalyE-Mail: [email protected]URL: russodipmatunina.altervista.org Received by the editors: 25.02.2010and in final form 25.02.2010.