Anti-CC-Groups and Anti-PC-Groups

A group G has Černikov classes of conjugate subgroups if the quotient group G/ coreG(NG(H)) is a Černikov group for each subgroup H of G. An anti-CC-group G is a group in which each nonfinitely generated subgroup K has the quotient group G/ coreG(NG(K)) which is a Černikov group. Analogously, a group G has polycyclic-by-finite classes of conjugate subgroups if the quotient group G/coreG(NG(H)) is a polycyclic -byfinite group for each subgroup H of G. An anti-PC-group G is a group in which each nonfinitely generated subgroup K has the quotient group G/coreG(NG(K)) which is a polycyclic-by-finite group. Anti-CC-groups and anti-PC-groups are the subject of the present article.


Introduction
The groups in which each subgroup has only finitely many conjugates have been characterized by B. H. Neumann [ Neumann's theorem can be formulated in terms of classes of groups as follows.For a subgroup H of a group G, we write  [1,3] for details about Černikov groups).This formulation has been recently introduced in [2], obtaining a satisfactory description as testified in [2,Main Theorem].The initial work of Polovickiȋ [4] gave a description of a periodic group G with Černikov classes of conjugate subgroups by showing that G is central-by-Černikov, that is, G has G/Z(G) which is a Černikov group.Since the class of Černikov groups extends the class of finite groups, Neumann's theorem can be found as a special situation in [2,Proposition 2.4].
A second extension of the concept of group with finite classes of conjugate subgroups can be given as follows.A group G has polycyclic-by-finite classes of conjugate subgroups if G/core G (N G (H)) is a polycyclic-by-finite group for each subgroup H of G (see [1,5] for details about polycyclic-by-finite groups).This formulation has been recently introduced in [6], obtaining a satisfactory description as testified in [6,Main Theorem].Initially, [7, Theorem 5.5] describes a group G which is central-by-(polycyclic-by-finite), that is, G has G/Z(G) which is a polycyclic-by-finite group.References [7,Theorem 5.5] and [6,Main Theorem] allow us to see Neumann's theorem as a special situation.
Let χ be a property of subgroups in groups, and let L be a family of subgroups of a given group G.It is a long standing line of research in Group Theory to study those groups in which all subgroups belonging to the family L of subgroups have the property χ.The beginnings of this line reach back to works of Dedekind [8] and Miller and Moreno [9].
Examples of families of subgroups considered so far are the family L 1 of all proper subgroups, the family L 2 of all finite subgroups, L 3 of all infinite subgroups, L 4 of all abelian subgroups, L 5 of all nonabelian subgroups, and L 6 of all finitely generated subgroups; while subgroups properties considered are for instance to be normal, subnormal, and subnormal of bounded defect, complemented, supplemented, and almost normal, or to satisfy min, max, min-∞, and max-∞ (see [1,10] for details).The references [11][12][13][14][15][16][17][18][19][20][21] show part of the literature which has been devoted to this topic during the last years.
We can often obtain a fairly good description of the group G if the family L is not too distant from L 1 .If, on the other hand, L is not a small subfamily of L 1 , the information Francesco Russo 3 "all subgroups of G belonging to L have property χ" is rather restricted.We take the family L 6 as an example: the descriptions of groups, all of which finitely generated subgroups are subnormal (Baer-groups, see [1, Lemmas 2.34, 2.35]), almost normal (FC-groups, see [1]), or satisfying max (locally noetherian groups, see [1]), are rather unsatisfactory.An exception is the class of all groups, all of which finitely generated subgroups are normal.These are the Dedekind groups and they have been classified.Therefore it may be interesting to study groups in which a property χ is imposed on a large family of subgroups, for instance, on the family L 7 of all nonfinitely generated subgroups.Clearly, L 7 = L 1 /L 6 .For the property χ, we choose to have Černikov classes of conjugate subgroups.
So this article is devoted to groups G, satisfying either of the following properties: ) is a polycyclic-by-finite group.
A group G which satisfies (i) is called anti-CC-group in analogy with the terminology which has been adopted in [13], where anti-FC-groups have been analyzed which is a finite group.Therefore, the notions of the anti-CC-group and anti-PC-group extend the notion of the anti-FC-group so that most of the results in [13] can be found as special situations.Section 2 is devoted to recall some preliminaries which help us to prove the main results.Our main results are contained in Sections 3 and 4.More precisely, Section 3 describes locally finite anti-CC-groups and anti-PC-groups.Section 4 describes locally nilpotent anti-CC-groups and anti-PC-groups.
Our notation is standard and can be found in [1].The background has been referred to [1,Section 4.3] for FC-groups, to [4,22,23] for CC-groups, and to [7] for PC-groups.General information on locally finite and locally nilpotent groups can be found in [10,14,24].

Preliminary results
Let G be a group.An element [25,Section 3]).In a similar way, an element The next result overlaps [25,Lemma 3.2] and it is shown only to the convenience of the reader.
Lemma 2.1.Let G be a group and let n be a positive integer.
(i) G is an FC-group if and only if Then each finitely generated subgroup of G is almost normal in G and this implies that G is an FC-group.Then (i) has been proved.A similar argument shows (ii) and (iii).
We end this section, recalling two results which are fundamental in our investigations.The first result describes the structure of a group with Černikov classes of conjugate subgroups (see [2, Main Theorem], [4]).
Theorem 2.2 [2].Let G be a group with Černikov classes of conjugate subgroups.Then the following assertions hold: Francesco Russo 5 A group G which has an abelian normal subgroup A such that G/A is a Černikov group and is said to be abelian-by-Černikov.This situation happens in statement (i) of the preceding theorem.
The second result describes the structure of a group with polycyclic-by-finite classes of conjugate subgroups [6, Main Theorem].
Theorem 2.3 [6].A group G has polycyclic-by-finite classes of a conjugate subgroups if and only if it is central-by-(polycyclic-by-finite).

Locally finite case
The first two statements follow from the definitions and from Lemma 2.1, so the proofs have been omitted.Proof.Consider x 1 = x ∩ A. Then supp x 1 = I 1 is a finite subset of I, and x ∩ Dr i∈M A i = 1, where M = I \ I 1 is infinite.We choose two infinite subsets M 1 and ).We note that is isomorphic to thanks to the well-known results of isomorphism between groups.G/K 1 ∩ K 2 is a Černikov group because it is the subdirect product of the Černikov groups G/K 1 and G/K 2 .Then G/core G (N G ( x )) is a Černikov group, and so x belongs to C(G).

Lemma 3.4. Assume that x is an element of the anti-PC-group G. If A = Dr i∈I A i is a subgroup of G consisting of x -invariant nontrivial direct factors A i , i ∈ I, with infinite index set I, then x belongs to P(G).
Proof.We follow the argument of the previous proof, using polycyclic-by-finite groups instead of Černikov groups.Proof.We define two subsets of I, namely, for every n ∈ N}.Obviously, M 1 ∪ M 2 = I, so at least one of the two subsets is infinite.
Now A = Dr i∈I x −r Ax r for every positive integer r, where x is an element of G and we obtain that is a direct product of infinitely many nontrivial factors since γ n (A i ) ≤ T. By construction, x normalizes T and permutes the given direct factors of T. By combining the conjugates under x to one new factor, we have reduced the situation to that of Lemma 3.3, and find that x belongs to C(G).
Case 2 (M 1 is infinite).Then the abelian group Z(A) is normalized by x and centralized by x n .Clearly, Z(A) is of infinite rank.Denote by W the torsion subgroup of Z(A).Again W is normalized by x.If the set of primes π occurring as orders of elements of W is infinite, we may define two subsets π 1 , π 2 of π, both infinite such that π 1 ∪ π 2 = π and If M 1 is infinite and the torsion subgroup W is of a infinite rank but π is finite, there is a characteristic elementary abelian p-subgroup V of W which is of infinite rank.Again, V is the direct product of two infinite x -invariant subgroups V 1 and If the torsion subgroup W is of finite rank, we can construct a torsion-free x -invariant subgroup L of infinite rank in Z(A).Again, x -invariant subgroups of infinite rank L 1 , L 2 can be chosen with L 1 ∩ x L 2 = 1, and L 2 L 1 = L. Now x L 1 , x L 2 , and x L 1 ∩ x L 2 = x belong to C(G).This completes Case 2, and the result follows.

Francesco Russo 7
Lemma 3.8.Assume that g is an element of the anti-PC-group G and A = Dr i∈I A i is a subgroup of G, with I as in Lemma 3.4.If g ∈ N G (A) and g n ∈ C G (A) for some positive integer n, then g belongs to P(G).
Proof.We follow the argument of the previous proof, using polycyclic-by-finite groups instead of Černikov groups and Lemma 3.4 instead of Lemma 3.3.Corollary 3.9.If the anti-CC-group G has an abelian torsion subgroup that does not satisfy the minimal condition on its subgroups, then all elements of finite order belong to C(G).
Proof.Denote the torsion subgroup of C(G) by T. We deduce from Corollary 3.5 that T does not satisfy min-ab.Choose an element x of finite order in G.A result of Zaȋtsev [21] implies that T possesses an abelian x -invariant subgroup A that does not satisfy min-ab.From Lemma 3.7, x belongs to C(G).

Corollary 3.10. If the anti-PC-group G has an abelian torsion subgroup that does not satisfy the minimal condition on its subgroups, then all elements of finite order belong to P(G).
Proof.We follow the argument of the previous proof, using polycyclic-by-finite groups instead of Černikov groups, Corollary 3.6 instead of Corollary 3.5, and Lemma 3.8 instead of Lemma 3.7.Proof.From Theorem 3.11, either G has Černikov classes of conjugate subgroups or G is a Černikov group.In the first case, we may apply (iv) of Theorem 2.2 so that the result follows.
Corollary 3.14.If G is a locally finite anti-PC-group, then either G is central-by-finite or G is a Černikov group.
Proof.From Theorem 3.12, either G has finite classes of conjugate subgroups or G is a Černikov group.In the first case, we recall that this is a different formulation of the Neumann's theorem, as mentioned in the introduction of the present paper.Then the result follows.
It seems opportune to note that Theorems 3.11 and 3.12 include [13, Theorem 2.2] as a special case, and agree with [15,Theorem 1].Now the classification of the locally finite anti-CC-group is easy to see.Proof.If G is not a Černikov group, then the result follows from Corollary 3.14.

Locally nilpotent case
A group G is called soluble-by-f inite if it has a normal soluble subgroup S whose index |G : S| is finite.We recall that a group G has finite abelian section rank if it has no infinite elementary abelian p sections for every prime p (see [1, volume II, Section 10]).Following [1,13], a soluble-by-finite group G is an 1 -group if it has finite abelian section rank and the set of prime divisors of orders of elements of G is finite.Literature on 1groups can be found, for instance, in [1,volume II].Finally, we recall the notion of rank of a group, following the well-known terminology of Prüfer (see [1]).If A is an abelian group, the torsion-free rank of A is the rank of the factor group A/T(A), where T(A) denotes the set of all elements of finite order in A. The torsion-free rank of A is denoted by r 0 (A).The total rank of A is the sum r 0 (A) + p r p (A), where r p (A) is the rank of the p components of A for each prime number p. has finite total rank by Corollary 3.5.A result of Charin (see [1,Theorem 6.36]) implies that K is a soluble 1 -group.We conclude that G has a normal soluble 1 -subgroup K such that G/K is a Černikov group.Therefore, G is an extension of a soluble 1 -group by an abelian group with min by a finite group.An abelian group with min is clearly an 1 -group and the class of 1 -groups is closed with respect to extensions of two of its members (see [1,15]).Therefore, G is a soluble-by-finite 1 -group.Theorem 4.2.Let G be an anti-PC-group having an ascending series whose factors are either locally nilpotent or locally finite.Then G has finite classes of conjugate subgroups or is a soluble-by-finite 1 -group.
Proof.We repeat the argument of the previous proof so that it is shown only for the convenience of the reader.
G possesses an ascending normal series whose factors are either locally nilpotent or locally finite [1,Theorem 2.31].Let K be the largest radical normal subgroup of G.It follows from Corollary 3.14 that the largest locally finite normal subgroup T/K of G/K is either central-by-finite or a Černikov group.From then, we repeat exactly the corresponding part in the proof of Theorem 4.1, using Corollary 3.6 instead of Corollary 3.5.It follows that G is a soluble-by-finite 1 -group.Proof.This follows from Theorem 4.2 and the formulation of Neumann's theorem as in the introduction.
It is well known that a locally nilpotent group G has its torsion subgroup T which is locally finite and the quotient group G/T which is torsion-free (see [1]).Then it is enough to investigate the structure of a torsion-free locally nilpotent anti-CC-group (resp., anti-PC-group) in order to have a satisfactory description of a locally nilpotent anti-CC-group (resp., anti-PC-group).[G,G] should be a Černikov group from Theorem 2.2 and this cannot be.Then we may assume that G is a soluble-by-finite 1 -group, since G is nonfinitely generated, also its center Z(G) is nonfinitely generated from [27,Lemma 2.6].Let X/Z(G) be a subgroup of G/Z(G).Then X is nonfinitely generated, and hence G/core G (N G (X)) is a Černikov group.But every subgroup of G/Z(G) has such property so that G/Z(G) has Černikov classes of conjugate subgroups.Now G/Z(G) satisfies Theorem 2.2 so that its derived

Lemma 3 .Lemma 3 . 3 .
1. (i) Subgroups and quotient groups of anti-CC-groups are anti-CC-groups.(ii)Subgroups and quotient groups of anti-PC-groups are anti-PC-groups.Lemma 3.2.(i) If G is an anti-CC-group and C(G) = G, then G has Černikov classes of conjugate subgroups.(ii)If G is an anti-PC-group and P(G) = G, then G has polycyclic-by-finite classes of conjugate subgroups.Assume that x is an element of the anti-CC-group G.If A = Dr i∈I A i is a subgroup of G consisting of x -invariant nontrivial direct factors A i , i ∈ I, with infinite index set I, then x belongs to C(G).

Corollary 3 . 5 .Corollary 3 . 6 .Lemma 3 . 7 .
Let G be an anti-CC-group and A = Dr i∈I A i a subgroup of G consisting of infinitely many nontrivial direct factors.Then A is contained in C(G).Let G be an anti-PC-group and A = Dr i∈I A i a subgroup of G consisting of infinitely many nontrivial direct factors.Then A is contained in P(G).Assume that g is an element of the anti-CC-group G and A = Dr i∈I A i is a subgroup of G, with I as in Lemma 3.3.If g ∈ N G (A) and g n ∈ C G (A) for some positive integer n, then g belongs to C(G).

Theorem 3 . 11 .
If G is a locally finite anti-CC-group, then either G has Černikov classes of conjugate subgroups or G is a Černikov group.Proof.If G does not satisfy min-ab, then G = C(G) by Corollary 3.9.From Lemma 3.2, G has Černikov classes of conjugate subgroups.If G satisfies min-ab, then a famous result of Shunkov [1, page 98] implies that G is a Černikov group.Theorem 3.12.If G is a locally finite anti-PC-group, then either G has finite classes of conjugate subgroups or G is a Černikov group.Proof.If G does not satisfy min-ab, then G = P(G) by Corollary 3.10.From Lemma 3.2, G has polycyclic-by-finite classes of conjugate subgroups.Then Theorem 2.3 implies that G/Z(G) is a polycyclic-by-finite group.Since G is periodic, G/Z(G) is a finite group.If G satisfies min-ab, then a famous result of Shunkov [1, page 98] implies that G is a Černikov group.Corollary 3.13.If G is a locally finite anti-CC-group, then either G is central-by-Černikov or G is a Černikov group.

Corollary 4 . 3 . 2 . 4 . 4 .
Let G be an anti-CC-group having an ascending series whose factors are either locally nilpotent or locally finite.Then G is abelian-by-Černikov or a soluble-by-finite1 -group.Proof.This follows from Theorems 4.1 and 2.Corollary Let G be an anti-PC-group having an ascending series whose factors are either locally nilpotent or locally finite.Then G is central-by-finite or a soluble-by-finite 1group.

Proposition 4 . 5 .
Let G be a torsion-free locally nilpotent anti-CC-group.If G is neither finitely generated nor abelian, then it is nilpotent of class 2.Proof.Assume from Theorem 4.1 that G has Černikov classes of conjugate subgroups.
1, Section 4, page 127] more than fifty years ago.A group G which has the center Z(G) of finite index in G is called central-by-finite.B. H. Neumann showed that a group is central-by-finite if and only if each subgroup has only finitely many conjugates.A subgroup H of a group G is called almost normal in G if H has finitely many conjugates in G, that is, if H has finite index |G : N G (H)|, where N G (H) is the normalizer of H in G. Therefore, Neumann's theorem [1, Section 4, page 127] shows that a centralby-finite group is characterized to have each subgroup, which is almost normal.
.1) where Cl G (H) denotes the set of conjugates of H in G. Clearly, core G (N G (H)) is a normal subgroup of G and The index |G : N G (H)| = |Cl G (H)| is finite if and only if the quotient group G/ core G (N G (H)) is finite.We will say that G has finite classes of conjugate subgroups if G/ core G (N G (H)) is a finite group for each subgroup H of G. Thus Neumann's theorem asserts that a group G has G/core G (N G (H)), which is a finite group for each subgroup H of G if and only if G is central-by-finite [2, Introduction].It is clear that H is almost normal in G if and only if G/core G (N G (H)) is a finite group.A first extension of the concept of group with finite classes of conjugate subgroups can be given as follows.A group G has Černikov finite classes of conjugate subgroups if G/core G (N G (H)) is a Černikov group for each subgroup H of G (see . An anti-FCgroup G is a group in which each nonfinitely generated subgroup H is almost normal in G.A group G which satisfies (ii) is called anti-PC-group.From the previous considerations, it is clear that a group G is an anti-FC-group if and only if each nonfinitely generated subgroup Theorem 3.15.The infinite locally finite group G which is not a Černikov group is an anti-CC-group if and only if G is central-by-Černikov.Proof.If G is not a Černikov group, then the result follows from Corollary 3.13.In a similar way, the classification of the locally finite anti-PC-group is easy to see.Theorem 3.16.The infinite locally finite group G which is not a Černikov group is an anti-PC-group if and only if G is central-by-finite.
Theorem 4.1.Let G be an anti-CC-group having an ascending series whose factors are either locally nilpotent or locally finite.Then G has Černikov classes of conjugate subgroups or is a soluble-by-finite 1 -group.Proof.G possesses an ascending normal series whose factors are either locally nilpotent or locally finite [1, Theorem 2.31].Let K be the largest radical normal subgroup of G.It follows from Corollary 3.13 that the largest locally finite normal subgroup T/K of G/K is either central-by-Černikov or a Černikov group.On the other hand, the factor group G/K has no nontrivial locally nilpotent normal subgroups, and hence T/K is a Černikov group.If H/T is a locally nilpotent normal subgroup of G/T, then the centralizer C H/K (T/K) is a locally nilpotent normal subgroup of G/K so that C H/K (T/K) = 1 and H/K is a Černikov group.It follows that T = G so that G has a normal radical subgroup K such that T/K is a Černikov group (in this situation, G is said to be a radical-by-Černikov group).Assume that G has Černikov classes of conjugate subgroups.Then every abelian subgroup of G FrancescoRusso 9