## Emis.uhasselt.be

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2007, Article ID 29423, pagesdoi:10.1155/2007/29423

**Anti-***CC***-Groups and Anti-***PC***-Groups**
Received 8 October 2007; Accepted 15 November 2007
Cernikov classes of conjugate subgroups if the quotient group

*G/*
core

*G*(

*NG*(

*H*)) is a ˇCernikov group for each subgroup

*H *of

*G*. An anti-

*CC*-group

*G *isa group in which each nonfinitely generated subgroup

*K *has the quotient group

*G/*core

*G*(

*NG*(

*K*)) which is a ˇCernikov group. Analogously, a group

*G *has polycyclic-by-finiteclasses of conjugate subgroups if the quotient group

*G/*core

*G*(

*NG*(

*H*)) is a polycyclic -by-finite group for each subgroup

*H *of

*G*. An anti-

*PC*-group

*G *is a group in which eachnonfinitely generated subgroup

*K *has the quotient group

*G/*core

*G*(

*NG*(

*K*)) which is apolycyclic-by-finite group. Anti-

*CC*-groups and anti-

*PC*-groups are the subject of thepresent article.

Copyright 2007 Francesco Russo. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

**1. Introduction**
The groups in which each subgroup has only finitely many conjugates have been charac-terized by B. H. Neumann 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. Neumannshowed that a group is

*central*-

*by*-

*finite *if and only if each subgroup has only finitely manyconjugates. A subgroup

*H *of a group

*G *is called

*almost normal *in

*G *if

*H *has finitely manyconjugates in

*G*, that is, if

*H *has finite index

*|G *:

*NG*(

*H*)

*|*, where

*NG*(

*H*) is the normalizerof

*H *in

*G*. Therefore, Neumann’s theorem Section 4, page 127] shows that a central-by-finite group is characterized to have each subgroup, which is almost normal.

Neumann’s theorem can be formulated in terms of classes of groups as follows. For a
subgroup

*H *of a group

*G*, we write
International Journal of Mathematics and Mathematical Sciences

*NG ClG*(

*H*)

*= *core

*G NG*(

*H*)

*=*
where

*ClG*(

*H*) denotes the set of conjugates of

*H *in

*G*. Clearly, core

*G*(

*NG*(

*H*)) is a normalsubgroup of

*G *and
The index

*|G *:

*NG*(

*H*)

*| = |ClG*(

*H*)

*| *is finite if and only if the quotient group

*G/*core

*G*(

*NG*(

*H*)) is finite. We will say that

*G *has

*finite classes of conjugate subgroups *if

*G/*core

*G*(

*NG*(

*H*)) is a finite group for each subgroup

*H *of

*G*. Thus Neumann’s theorem as-serts that a group

*G *has

*G/*core

*G*(

*NG*(

*H*)), which is a finite group for each subgroup

*H *of

*G *if and only if

*G *is central-by-finite Introduction]. It is clear that

*H *is almost normalin

*G *if and only if

*G/*core

*G*(

*NG*(

*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

*ˇ*
*Cernikov finite classes of conjugate subgroups *if

*G/*core

*G*(

*NG*(

*H*)) is a ˇCernikov group for each subgroup

*H *of

*G *(see for detailsabout ˇ
Cernikov groups). This formulation has been recently introduced in ], obtaining
a satisfactory description as testified in , Main Theorem]. The initial work of Polovicki˘ı] gave a description of a periodic group

*G *with ˇ
by showing that

*G *is

*central-by- ˇ*
*Cernikov*, that is,

*G *has

*G/Z*(

*G*) which is a ˇ
Cernikov groups extends the class of finite groups, Neumann’s theorem
can be found as a special situation in 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*(

*NG*(

*H*)) is a polycyclic-by-finite group for each subgroup

*H *of

*G *(see fordetails about polycyclic-by-finite groups). This formulation has been recently introducedin ], obtaining a satisfactory description as testified in , Main Theorem]. Initially, ,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 Theorem 5.5] and MainTheorem] 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 inwhich all subgroups belonging to the family

*L *of subgroups have the property

*χ*. Thebeginnings of this line reach back to works of Dedekind ] and Miller and Moreno
Examples of families of subgroups considered so far are the family

*L *of all proper
subgroups, the family

*L *of all finite subgroups,

*L *of all infinite subgroups,

*L *of all
abelian subgroups,

*L *of all nonabelian subgroups, and

*L *of all finitely generated sub-
groups; while subgroups properties considered are for instance to be normal, subnor-mal, and subnormal of bounded defect, complemented, supplemented, and almost nor-mal, or to satisfy min, max, min-

*∞*, and max-

*∞ *(see ] for details). The referencesshow part of the literature which has been devoted to this topic during the lastyears.

We can often obtain a fairly good description of the group

*G *if the family

*L *is not too
distant from

*L *. If, on the other hand,

*L *is not a small subfamily of

*L *, the information
“all subgroups of

*G *belonging to

*L *have property

*χ*” is rather restricted. We take the fam-ily

*L *as an example: the descriptions of groups, all of which finitely generated subgroups
are subnormal (Baer-groups, see , Lemmas 2.34, 2.35]), almost normal (

*FC*-groups,see ]), or satisfying max (locally noetherian groups, see are rather unsatisfactory.

An exception is the class of all groups, all of which finitely generated subgroups are nor-mal. These are the Dedekind groups and they have been classified. Therefore it may beinteresting to study groups in which a property

*χ *is imposed on a

*large *family of sub-groups, for instance, on the family

*L *of all nonfinitely generated subgroups. Clearly,

*L = L /L *. For the property

*χ*, we choose to have ˇCernikov classes of conjugate sub-
So this article is devoted to groups

*G*, satisfying either of the following properties:
(i) if the subgroup

*H *of

*G *is nonfinitely generated,
then

*G/*core

*G*(

*NG*(

*H*)) is a ˇCernikov group;
(ii) if the subgroup

*H *of

*G *is nonfinitely generated,
then

*G/*core

*G*(

*NG*(

*H*)) is a polycyclic-by-finite group.

A group

*G *which satisfies (i) is called anti-

*CC*-group in analogy with the terminologywhich has been adopted in where anti-

*FC*-groups have been analyzed. An anti-

*FC*-group

*G *is a group in which each nonfinitely generated subgroup

*H *is almost normalin

*G*. A group

*G *which satisfies (ii) is called anti-

*PC*-group. From the previous consid-erations, it is clear that a group

*G *is an anti-

*FC*-group if and only if each nonfinitelygenerated subgroup

*H *of

*G *has

*G/*core

*G*(

*NG*(

*H*)) which is a finite group. Therefore, thenotions of the anti-

*CC*-group and anti-

*PC*-group extend the notion of the anti-

*FC*-groupso that most of the results in ] can be found as special situations.

is devoted to recall some preliminaries which help us to prove the main
results. Our main results are contained in Sections and More precisely, describes locally finite anti-

*CC*-groups and anti-

*PC*-groups. describes locallynilpotent anti-

*CC*-groups and anti-

*PC*-groups.

Our notation is standard and can be found in The background has been referred
to Section 4.3] for

*FC*-groups, to , for

*CC*-groups, and to ] for

*PC*-groups.

General information on locally finite and locally nilpotent groups can be found in ,

**2. Preliminary results**
Let

*G *be a group. An element

*x *of

*G *is called

*FC*-

*element *of

*G *if

*G/CG*(

*x G*) is a finitegroup. The set

*F*(

*G*) of all

*FC*-elements of

*G *is a characteristic subgroup of

*G*, which iscalled

*FC*-

*center *of

*G *, Section 4.3]. In a similar way, an element

*x *of

*G *is called

*CC*-element of

*G *if

*G/CG*(

*x G*) is a ˇCernikov group. The set

*C*(

*G*) of all

*CC*-elements of

*G *isa characteristic subgroup of

*G*, which is called

*CC*-

*center *of

*G *(see Section 3]). In asimilar way, an element

*x *of

*G *is called

*PC*-element of

*G *if

*G/CG*(

*x G*) is a polycyclic-by-finite group. The set

*P*(

*G*) of all

*PC*-elements of

*G *is a characteristic subgroup of

*G*, whichis called

*PC*-

*center *of

*G *(see Obviously,

*G *is an

*FC*-group if and only if

*G = F*(

*G*).

Similarly,

*G *is a

*CC*-group if and only if

*G = C*(

*G*). Similarly,

*G *is a

*PC*-group if and onlyif

*G = P*(

*G*).

International Journal of Mathematics and Mathematical Sciences
The next result overlaps Lemma 3.2] and it is shown only to the convenience of
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*
*F*(

*G*)

*= H = h*1,

*.*,

*hn *:

*G/*core

*G NG*(

*H*)

*is a finite group .*
(ii)

*G is a CC-group if and only if*
*C*(

*G*)

*= H = h*1,

*.*,

*hn *:

*G/*core

*G NG*(

*H*)

*is a ˇ*
(iii)

*G is a PC-group if and only if*
*P*(

*G*)

*= H = h*1,

*.*,

*hn *:

*G/*core

*G NG*(

*H*)

*is a polycyclic-by-finite-group .*
*Proof. *Assume that

*G *is an

*FC*-group,

*x *is an

*FC*-element of

*G *and

*K = H = h*1,

*.*,

*hn *:

*G/*core

*G*(

*NG*(

*H*)) is finite . If

*a ∈ CG*(

*x G*), then [

*by*,

*a*]

*= *1 for each

*b ∈ x *and

*y ∈ G*,in particular,
Therefore,

*CG*(

*x G*) is contained in core

*G*(

*NG*(

*x *)) so that

*G/*core

*G*(

*NG*(

*x *)) is a finitegroup and

*x *belongs to

*K*. Then

*F*(

*G*)

*≤ K*, but

*F*(

*G*)

*= G *so that

*G = K*. Conversely,assume that

*F*(

*G*)

*= K*. 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).

Reference describes those groups in which each nonfinitely generated subgroup is
subnormal. Such groups are called

*db*-groups and they represent the dual class of the Baergroups (see , Section 2.3]). Unfortunately, we cannot say that an anti-

*CC*-group(resp., an anti-

*PC*-group) is a

*db*-group so that many results of ] cannot be directlyapplied. However, it is possible to compare Theorems 2.2, 2.11, 2.13, 3.6, 3.11, 3.16,3.17, 4.6, 4.8, 4.11, 4.12, 4.15, 4.16] with Theorems 1, 2, 3, 4, 5], noting that analogoussituations happen for anti-

*CC*-groups (resp., for anti-

*PC*-groups). In particular, somemethods which have been used in the present paper mime the methods which have beenused in , ].

We end this section, recalling two results which are fundamental in our investigations.

The first result describes the structure of a group with ˇ
Theorem 2.2

*Let G be a group with ˇ*
*Cernikov classes of conjugate subgroups. Then the*
(i)

*G has an abelian normal subgroup A such that G/A is a ˇ*
(ii)

*if T is the torsion subgroup of A, then G/CG*(

*T*)

*is a finite group;*
(iii) [

*G*,

*G*]

*is a ˇ*
(iv)

*if G is periodic, then G is a central-by- ˇ*
A group

*G *which has an abelian normal subgroup

*A *such that

*G/A *is a ˇ
group and is said to be

*abelian*-

*by*-

*ˇ*
*Cernikov*. This situation happens in statement (i) of
The second result describes the structure of a group with polycyclic-by-finite classes of
conjugate subgroups , Main Theorem].

Theorem 2.3

*A group G has polycyclic-by-finite classes of a conjugate subgroups if andonly if it is central-by-(polycyclic-by-finite).*
**3. Locally finite case**
The first two statements follow from the definitions and from so the proofshave been omitted.

Lemma 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 ˇ*
(ii)

*If G is an anti-PC-group and P*(

*G*)

*= G, then G has polycyclic-by-finite classes of*
Lemma 3.3.

*Assume that x is an element of the anti-CC-group G. If A = Dri∈IAi is a sub-group of G consisting of x -invariant nontrivial direct factors Ai, i ∈ I, with infinite indexset I, then x belongs to C*(

*G*)

*.*
*Proof. *Consider

*x*1

*= x ∩ A*. Then supp

*x*1

*= I*1 is a finite subset of

*I*, and

*x ∩Dri∈MAi = *1, where

*M = I \ I*1 is infinite. We choose two infinite subsets

*M*1 and

*M*2of

*M *such that

*M*1

*∪ M*2

*= M *and

*M*1

*∩ M*2

*= *∅. Obviously,

*H*1

*= x Dri∈M A*
*i *cannot be finitely generated, therefore,

*G/*core

*G*(

*NG*(

*H*1 )) and

*G/*core

*G*(

*NG*(

*H*2 )) are ˇCernikov groups. Put

*K*1

*= *core

*G*(

*NG*(

*H*1 )) and

*K*2

*=*core

*G*(

*NG*(

*H*2 )). We note that

*K*1

*∩ K*2

*≤ *core

*G NG H*1

*∩ H*2

*G/*core

*G NG H*1

*∩ H*2
thanks to the well-known results of isomorphism between groups.

*G/K*1

*∩ K*2 is a
Cernikov group because it is the subdirect product of the ˇ

*G/K*2. Then

*G/*core

*G*(

*NG*(

*x *)) is a ˇCernikov group, and so

*x *belongs to

*C*(

*G*).

International Journal of Mathematics and Mathematical Sciences
Lemma 3.4.

*Assume that x is an element of the anti-PC-group G. If A = Dri∈IAi is a sub-group of G consisting of x -invariant nontrivial direct factors Ai, i ∈ I, with infinite indexset I, then x belongs to P*(

*G*)

*.*
*Proof. *We follow the argument of the previous proof, using polycyclic-by-finite groupsinstead of ˇ
Corollary 3.5.

*Let G be an anti-CC-group and A = Dri∈IAi a subgroup of G consisting ofinfinitely many nontrivial direct factors. Then A is contained in C*(

*G*)

*.*
Corollary 3.6.

*Let G be an anti-PC-group and A = Dri∈IAi a subgroup of G consisting ofinfinitely many nontrivial direct factors. Then A is contained in P*(

*G*)

*.*
Lemma 3.7.

*Assume that g is an element of the anti-CC-group G and A = Dri∈IAi is asubgroup of G, with I as in **If g ∈ NG*(

*A*)

*and gn ∈ CG*(

*A*)

*for some positiveinteger n, then g belongs to C*(

*G*)

*.*
*Proof. *We define two subsets of

*I*, namely,

*M*1

*= {i *:

*Z*(

*Ai*)

*=*1

*} *and

*M*2

*= {i *:

*γ *(

*A*
for every

*n ∈ *N

*}. *Obviously,

*M*1

*∪ M*2

*= I*, so at least one of the two subsets is infinite.

*Case 1 (M*2

*is infinite). *If

*D*1,

*.*,

*Dn *are normal subgroups of a group

*F*, then [

*.*[[

*D*1,

*D*2],

*D*3],

*.*,

*Dn*] is a normal subgroup of

*F*, which is contained in

*ni=*1

*Di*, furthermore, [

*Di*,

*DjDk*]

*= *[

*Di*,

*Dj*][

*Di*,

*Dk*].

Now

*A = Dri∈Ix−rAxr *for every positive integer

*r*, where

*x *is an element of

*G *and we

*∩ x−*1

*A x ∩ x−*2

*A x*2

*∩ ··· ∩ x−n*+1

*A xn−*1
is a direct product of infinitely many nontrivial factors since

*γ *(

*A*
*i*)

*≤ T *. By construction,

*x *normalizes

*T *and permutes the given direct factors of

*T*. By combining the conjugatesunder

*x *to one new factor, we have reduced the situation to that of and findthat

*x *belongs to

*C*(

*G*).

*Case 2 (M*1

*is infinite). *Then the abelian group

*Z*(

*A*) is normalized by

*x *and centralizedby

*xn*. 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 *isinfinite, we may define two subsets

*π*1,

*π*2 of

*π*, both infinite such that

*π*1

*∪ π*2

*= π *and

*π*1

*∩ π*2

*= *∅. If

*W*1 and

*W*2 are the corresponding

*π j*-Sylow subgroups of

*W *(

*j = *1,2),then

*x W*1,

*x W*2, and

*x W*1

*∩ x W*2

*= x *belong to

*C*(

*G*).

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

*V*2 such that

*V*1

*∩*
*x V*2

*= *1. Again,

*x V*1,

*x V*2, and

*x V*1

*∩ x V*2

*= x *belong to

*C*(

*G*). 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 bechosen 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,
Lemma 3.8.

*Assume that g is an element of the anti-PC-group G and A = Dri∈IAi is asubgroup of G, with I as in **If g ∈ NG*(

*A*)

*and gn ∈ CG*(

*A*)

*for some positiveinteger n, then g belongs to P*(

*G*)

*.*
*Proof. *We follow the argument of the previous proof, using polycyclic-by-finite groupsinstead of ˇ
Corollary 3.9.

*If the anti-CC-group G has an abelian torsion subgroup that does notsatisfy the minimal condition on its subgroups, then all elements of finite order belong toC*(

*G*)

*.*
*Proof. *Denote the torsion subgroup of

*C*(

*G*) by

*T*. We deduce from that

*T *does not satisfy min-

*ab*. Choose an element

*x *of finite order in

*G*. A result of Za˘ıtsev] implies that

*T *possesses an abelian

*x *-invariant subgroup

*A *that does not satisfymin-

*ab*. From

*x *belongs to

*C*(

*G*).

Corollary 3.10.

*If the anti-PC-group G has an abelian torsion subgroup that does notsatisfy the minimal condition on its subgroups, then all elements of finite order belong toP*(

*G*)

*.*
*Proof. *We follow the argument of the previous proof, using polycyclic-by-finite groupsinstead of ˇ
Theorem 3.11.

*If G is a locally finite anti-CC-group, then either G has ˇ*
*Proof. *If

*G *does not satisfy min-

*ab*, then

*G = C*(

*G*) by From

*G*has ˇ
Cernikov classes of conjugate subgroups. If

*G *satisfies min-

*ab*, then a famous result
of Shunkov , page 98] implies that

*G *is a ˇ
Theorem 3.12.

*If G is a locally finite anti-PC-group, then either G has finite classes ofconjugate subgroups or G is a ˇ*
*Proof. *If

*G *does not satisfy min-

*ab*, then

*G = P*(

*G*) by From

*G *has polycyclic-by-finite classes of conjugate subgroups. Then 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 page 98] implies that

*G *is a ˇ
Corollary 3.13.

*If G is a locally finite anti-CC-group, then either G is central-by-ˇ*
*Proof. *From either

*G *has ˇ
Cernikov classes of conjugate subgroups or

*G *is
Cernikov group. In the first case, we may apply (iv) of so that the result
Corollary 3.14.

*If G is a locally finite anti-PC-group, then either G is central-by-finite orG is a ˇ*
International Journal of Mathematics and Mathematical Sciences

*Proof. *From either

*G *has finite classes of conjugate subgroups or

*G *is a
Cernikov group. In the first case, we recall that this is a diﬀerent formulation of the Neu-mann’s theorem, as mentioned in the introduction of the present paper. Then the resultfollows.

It seems opportune to note that Theorems and include as
a special case, and agree with Theorem 1].

Now the classification of the locally finite anti-

*CC*-group is easy to see.

Theorem 3.15.

*The infinite locally finite group G which is not a ˇ*
*CC-group if and only if G is central-by- ˇ*
Cernikov group, then the result follows from
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 ˇ*
*PC-group if and only if G is central-by-finite.*
Cernikov group, then the result follows from

**4. 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 infiniteelementary abelian

*p *sections for every prime

*p *(see volume II, Section 10]). Fol-lowing ], a soluble-by-finite group

*G *is an 1-group if it has finite abelian sectionrank and the set of prime divisors of orders of elements of

*G *is finite. Literature on 1-groups can be found, for instance, in , volume II]. Finally, we recall the notion of rankof a group, following the well-known terminology of Pr¨ufer (see ]). If

*A *is an abeliangroup, the

*torsion*-

*free rank *of

*A *is the rank of the factor group

*A/T*(

*A*), where

*T*(

*A*) de-notes 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*) +

*prp*(

*A*), where

*rp*(

*A*) is the

*rank *of the

*p*
components of

*A *for each prime number

*p*.

Theorem 4.1.

*Let G be an anti-CC-group having an ascending series whose factors areeither locally nilpotent or locally finite. Then G has ˇ*
*Cernikov classes of conjugate subgroups*
*or is a soluble-by-finite *1

*-group.*
*Proof. G *possesses an ascending normal series whose factors are either locally nilpotentor locally finite , Theorem 2.31]. Let

*K *be the largest radical normal subgroup of

*G*. Itfollows from that the largest locally finite normal subgroup

*T/K *of

*G/K *iseither central-by- ˇ
Cernikov group. On the other hand, the factor group

*G/K*
has no nontrivial locally nilpotent normal subgroups, and hence

*T/K *is a ˇ
If

*H/T *is a locally nilpotent normal subgroup of

*G/T*, then the centralizer

*CH/K *(

*T/K*) is alocally nilpotent normal subgroup of

*G/K *so that

*CH/K *(

*T/K*)

*= *1 and

*H/K *is a ˇCernikovgroup. It follows that

*T = G *so that

*G *has a normal radical subgroup

*K *such that

*T/K *isa ˇ
Cernikov group (in this situation,

*G *is said to be a radical-by- ˇ
Cernikov classes of conjugate subgroups. Then every abelian subgroup of

*G*
has finite total rank by A result of Charin (see Theorem 6.36]) impliesthat

*K *is a soluble 1-group. We conclude that

*G *has a normal soluble 1-subgroup

*K*such that

*G/K *is a ˇ
Cernikov 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 an1-group and the class of 1-groups is closed with respect to extensions of two of itsmembers (see , 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 areeither locally nilpotent or locally finite. Then G has finite classes of conjugate subgroups or isa soluble-by-finite *1

*-group.*
*Proof. *We repeat the argument of the previous proof so that it is shown only for theconvenience of the reader.

*G *possesses an ascending normal series whose factors are either locally nilpotent or
locally finite , Theorem 2.31]. Let

*K *be the largest radical normal subgroup of

*G*. Itfollows from that the largest locally finite normal subgroup

*T/K *of

*G/K*is either central-by-finite or a ˇ
Cernikov group. From then, we repeat exactly the corre-
sponding part in the proof of using instead of It follows that

*G *is a soluble-by-finite 1-group.

Corollary 4.3.

*Let G be an anti-CC-group having an ascending series whose factors areeither locally nilpotent or locally finite. Then G is abelian-by- ˇ*
*Proof. *This follows from Theorems and
Corollary 4.4.

*Let G be an anti-PC-group having an ascending series whose factors areeither locally nilpotent or locally finite. Then G is central-by-finite or a soluble-by-finite *1

*-group.*
*Proof. *This follows from and the formulation of Neumann’s theorem as inthe 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 Then it is enoughto 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).

Proposition 4.5.

*Let G be a torsion-free locally nilpotent anti-CC-group. If G is neitherfinitely generated nor abelian, then it is nilpotent of class 2.*
*Proof. *Assume from that

*G *has ˇ
Cernikov classes of conjugate subgroups.

Cernikov group from and this cannot be. Then we may
assume that

*G *is a soluble-by-finite 1-group, since

*G *is nonfinitely generated, also itscenter

*Z*(

*G*) is nonfinitely generated from Lemma 2.6]. Let

*X/Z*(

*G*) be a subgroupof

*G/Z*(

*G*). Then

*X *is nonfinitely generated, and hence

*G/*core

*G*(

*NG*(

*X*)) is a ˇCernikovgroup. But every subgroup of

*G/Z*(

*G*) has such property so that

*G/Z*(

*G*) has ˇ
classes of conjugate subgroups. Now

*G/Z*(

*G*) satisfies so that its derived
International Journal of Mathematics and Mathematical Sciences
subgroup [

*G/Z*(

*G*),

*G/Z*(

*G*)] is a ˇ
Cernikov group. We note that

*T*(

*G/Z*(

*G*))

*= T*(

*G*)

*Z*(

*G*)

*/*
*Z*(

*G*) and

*T*(

*G*)

*= *1, then

*T*(

*G/Z*(

*G*))

*= *1 and

*G/Z*(

*G*) is a torsion-free group. Now [

*G/Z*(

*G*),

*G/Z*(

*G*)]

*= *1 so that

*G/Z*(

*G*) is abelian, and

*G *is nilpotent of class 2.

Proposition 4.6.

*Let G be a torsion-free locally nilpotent anti-PC-group. If G is neitherfinitely generated nor abelian, then it is nilpotent of class 2.*
*Proof. *We may repeat the argument of the preceding proof, consider the correspondingstatements for anti-

*PC*-groups.

Theorem 4.7.

*Assume that G is a locally nilpotent anti-CC-group with torsion subgroup T.*

Then
(ii)

*G/T is torsion-free nilpotent of class 2, whenever it is neither finitely generated nor*
*Proof. *(i) follows from (ii) follows from
Theorem 4.8.

*Assume that G is a locally nilpotent anti-PC-group with torsion subgroup T.*

Then
(i)

*T is either central-by-finite or a ˇ*
(ii)

*G/T is torsion-free nilpotent of class 2, whenever it is neither finitely generated nor*
*Proof. *(i) follows from (ii) follows from

**5. Examples**
*Example 5.1. *Each anti-

*FC*-group is an anti-

*CC*-group as testified by definitions. Exam-ples of anti-

*FC*-groups can be found in , page 44, lines 1–13] or Example 3.12].

Of course, each anti-

*FC*-group is an anti-

*PC*-group.

*Example 5.2. *The Example which has been described in , Section 4] is a nonperiodicgroup with ˇ
Cernikov classes of conjugate subgroups. This example is an anti-CC-group.

Each central-by-(polycyclic-by-finite) group is an anti-

*PC*-group thanks to

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Francesco Russo: Department of Mathematics, Faculty of Mathematics, University of Naples,Via Cinthia, 80126 Naples, Italy

*Email address*:

**Journal of Applied Mathematics and Decision Sciences**
**Special Issue on**
**Decision Support for Intermodal Transport**
**Call for Papers**
Intermodal transport refers to the movement of goods in
Before submission authors should carefully read over the
a single loading unit which uses successive various modes
journal’s Author Guidelines, which are located at
of transport (road, rail, water) without handling the goods
during mode transfers. Intermodal transport has become
authors should submit an electronic copy of their complete
an important policy issue, mainly because it is considered
manuscript through the journal Manuscript Tracking Sys-
to be one of the means to lower the congestion caused by
single-mode road transport and to be more environmentally
friendly than the single-mode road transport. Both consider-ations have been followed by an increase in attention toward
intermodal freight transportation research.

Various intermodal freight transport decision problems
are in demand of mathematical models of supporting them.

As the intermodal transport system is more complex than asingle-mode system, this fact oﬀers interesting and challeng-ing opportunities to modelers in applied mathematics. This

**Lead Guest Editor**
special issue aims to fill in some gaps in the research agenda

**Gerrit K. Janssens, **Transportation Research Institute

of decision-making in intermodal transport.

(IMOB), Hasselt University, Agoralaan, Building D, 3590
The mathematical models may be of the optimization type
or of the evaluation type to gain an insight in intermodaloperations. The mathematical models aim to support deci-sions on the strategic, tactical, and operational levels. The

**Guest Editor**
decision-makers belong to the various players in the inter-

**Cathy Macharis, **Department of Mathematics, Operational

modal transport world, namely, drayage operators, terminal
Research, Statistics and Information for Systems (MOSI),
operators, network operators, or intermodal operators.

Transport and Logistics Research Group, Management
Topics of relevance to this type of decision-making both in
School, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel,
time horizon as in terms of operators are:
• Intermodal terminal design• Infrastructure network configuration• Location of terminals• Cooperation between drayage companies• Allocation of shippers/receivers to a terminal• Pricing strategies• Capacity levels of equipment and labour• Operational routines and lay-out structure• Redistribution of load units, railcars, barges, and so
• Scheduling of trips or jobs• Allocation of capacity to jobs• Loading orders• Selection of routing and service

Source: http://emis.uhasselt.be/journals/HOA/IJMMS/Volume2007/29423.pdf

DOI: 10.5379/urbani-izziv-en-2012-23-supplement-1-007 Knowledge Management in the Network Mode: The Case of Private Equity Britta Klagge, Institute of Geography, University of Osnabrück, Germany ([email protected]) Carsten Peter, Institute of Geography, University of Osnabrück, Germany ([email protected]) Abstract There has been an ongoing debate on the changing geographical organization o

Cardiovasc Intervent Radiol (2010) 33:11–17Quality Improvement Guidelines for RadiofrequencyAblation of Liver TumoursLaura Crocetti • Thierry de Baere •Riccardo LencioniReceived: 5 October 2009 / Accepted: 5 October 2009 / Published online: 19 November 2009 Ó Springer Science+Business Media, LLC and the Cardiovascular and Interventional Radiological Society of Europe (CIRSE) 2009The de