Groups with Prescribed Quotient Groups and Associated Module Theory

The following sections are included:

• Preface

• Contents

• Notation

As usual, a module A is said to be simple, if A does not include proper non-zero submodules.

As its title indicates, this part, formed by four chapters, is devoted to studying the existence and the structure of the simple modules over certain group rings, an important tool in the study of the structure of the groups in consideration.

Let R be an integral domain. If A is an R-module and a ∈ A, the annihilator of a is Ann_R(a) = {x ∈ R | ax = 0}, an ideal of R. The annihilator of A is

The following sections are included:

• Proposition

• Corollary

• Theorem [HM]

• Corollary

• Corollary

• Theorem

Having developed in full the abelian case, we are ready to deduce the structure of simple DG - modules where G is some generalization of abelian group. Nilpotent and hypercentral groups are the first natural extension of abelian groups. Other important generalizations of abelian groups are FC - groups and CC - groups (see definitions below). Thus, the description of simple modules over all these classes of groups will be the aims of this chapter…

Let A be a module over a ring R, B a submodule of A. We say that B has a complement in A (or B is a complemented submodule) if there is an R - submodule C such that A = B ⊕ C…

As we already noted, another type of modules which play a relevant role in the study of just non--groups are the just infinite modules. Because for some noetherian modules it is possible to make a reduction to just infinite modules, the investigation of just infinite modules is important by itself. These modules arise by the following way…

For the study of a module A over a group ring DG where D is a Dedekind domain the structure of A as a D-module plays a very important role. In this chapter we have collected some results about modules over Dedekind domains, which are necessary for the study of just infinite modules over DG. In particular, we consider an analogy of the important concept of a p - basic subgroup in Abelian Group Theory. We have no found analogies of this concept for modules over Dedekind domains in the journal literature. In this case, as well as in all remaining cases, we followed the analogy from ℤ to D. Our purpose is to prove the existence of P- basic submodules in torsion-free modules and, using this, to prove some results about torsion-free D-modules, which are necessary for us. Thus, this chapter hass a technical character…

Now, we are ready to start our study. Our first step is Theorem 6.6, allowing us “to change” the underlying ring of coefficients. The main part of this chapter is devoted to applications of this theorem to just infinite modules over FC - hypercentral groups. All these applications are based on reduction to a normal subgroup of finite index…

In the previous chapter we began to consider just infinite modules over an FC - hypercentral group G. As we have proved, in this case G is abelian-by-finite. However, we obtained virtually no information about the structure of these modules. Now we will consider the case in which G is a FC- hypercentral group with finite 0-rank…

Chronologically polycyclic-by-finite groups have formed the first class of groups, over which just infinite modules have been studied. D. J. S.Robinson and J. S. Wilson in their fundamental paper [RW] showed that the classification of just non-polycyclic groups heavily depends of the behavior of just infinite modules over polycyclic groups (even the term “just infinite module” has arisen in this work ). All subsequent papers devoted to various types of just non--groups showed that the study of just infinite modules is a necessary important component of this research…

Let D be a Dedekind domain.

A D- module A is called D - co-(layer-finite), or it is simply said that it is co-(layer-finite) as a D - module, if for every non-zero ideal I of D the factor-module A|AI is finitely generated D-module…

The set of all proper factor - groups of a group (i.e. the factor-groups by non-identity normal subgroups) is the widest family of group quotients. Let be a class of groups. A group G is said to be a just non- - group if every proper factor-group of G is an - group but G does not belong to . This part of the book is dedicated to just non- -groups for such natural classes of groups as abelian groups, finite groups and some their extensions, nilpotent and hypercentral groups, central-by-finite groups, groups with finite derived subgroups, FC - groups, CC - groups.

We will study such groups G with some additional conditions, which imply the non-simplicity of G. The following condition is typical for these purposes: Fitt(G) ≠ < 1 >, where Fitt(G) is the Fitting subgroup of G, that is the subgroup generated by all normal nilpotent subgroups. Naturally, the first step here is investigation of a structure of the Fitting subgroup of just non--groups. In fact, for groups from all indicated classes it is possible to prove that its Fitting subgroups are abelian. This allows us to consider them as modules over the group G/Fitt(G) already belonging to the class . The subsequent study is divided into two cases: the monolithic case and the non-monolithic case. In the monolithic case, as a rule, it is possible to prove, that Fitt(G) coincides with the monolith of a group; in the non-monolithic case Fitt(G) stands by a just infinite module. Therefore a tool in this study is the developing of properties of simple and just infinite modules, which were investigated in the previous chapters.

In this chapter, we collect some of the most relevant properties of just non--groups we need in the sequel, especially those which concern the Fitting subgroup…

Chronologically the class of just non-abelian groups was the first class of just non--groups. The structure of soluble such groups was determined by M. F. Newman in his papers [NM 1] and [NM 2]. Making use of the results previously showed in the precedent chapters of this book, we can carry out this study more concisely…

The class of all nilpotent groups and the class of all hypercentral groups are very natural extensions of the class of all abelian groups. Thus in our study of just non--groups, the consideration of just non-hypercentral groups and just non-nilpotent groups should be the next step. As in the previous chapters we are studying the just non-hypercentral groups G with Fitt(G) ≠ < 1 >. Some information about this subgroup was obtained in Chapter 10. In particular, A = Fitt(G) is abelian, and, as usually, we can consider A as a ℤH - module, where H = G/A is a hypercentral group. If B is a non-identity G - invariant subgroup( or, using the module language, B is a non-zero ℤH - submodule) of A, then G/B is hypercentral, in particular, a ℤH -factor-module A/B is ℤH - hypercentral…

In Chapter 12 we have proved the nilpotency of some groups (for example, soluble groups of finite 0 - rank and, in particular, periodic soluble groups), all proper factor-groups of which are nilpotent. In this connection the following problem comes on naturally. For what another family of proper factor-groups of a group G the nilpotency of each factor-group from the family needs the nilpotency of whole group G? The following result of K. Hirsch (see, for example, [SD 2, 1.C, Theorem 2]) gives us a major precedent of the consideration of this problem :

If every finite factor-group of a polycyclic group G is nilpotent, then G is nilpotent too…

Chronologically, just infinite groups (that is the just non-finite groups) were one of the first investigated classes of just non--groups. They are natural generalization of infinite cyclic groups. In fact, every non-identity subgroup of an infinite cyclic group has finite index. Moreover, the converse statement is also correct: if every non-identity subgroup of an infinite group has finite index, then this group is cyclic (Yu. G. Fedorov, [FYu]). The theory of just infinite groups has its own specificity. If the majority of results on just non--groups for other classes concern with generalized soluble groups, the development of the theory of just infinite groups requires to work with not only generalized soluble groups. As an illustration to that we can point the examples of just infinite finitely generated p-groups constructed by R. I. Grigorchuk [GRI 1, GRI 2] and N. Gupta, S. Sidki [GUS], the study of distinct properties of which will be carried out intensively now…

The classes of all polycyclic-by-finite groups and of all Chernikov groups are the most near extensions of the class of all finite groups and they have been studied intensively by several authors. Since these groups are finite extensions of soluble groups with the two main chain conditions Max and Min respectively (see[RD 9, 3.1]), the classes and are dual to each other in some sense. Also …

Now we want to consider just non--groups, where is a certain class of groups which includes both classes: the class of all finite groups and the class of all abelian groups. Specifically, we will consider the following candidates for : central-by-finite groups, finite-by-abelian groups, FC-groups, and CC-groups. Just non-(central-by-finite) and just non-(finite-by-abelian) groups have been investigated in the remarkable paper of D. J. S. Robinson and Z. Zheng [RZ ]. While in [RW ] the key condition on the respectively group ring was to be noetherian, in the case of central-by-finite or finite-by-abelian group H the group ring ℤH can not be noetherian. In this case the following fact is defining: the group H contains many almost central elements. Since central-by-finite and finite-by-abelian groups are contained in the wider class of FC - groups, the following natural step was the study of just non-FC-groups. This step has been realized by S. Franciosi, F. de Giovanni and L. A. Kurdachenko in [FdeGK 3]. In turn, the class of FC - groups is a subclass of the class of CC - groups. Therefore the study of just non-CC-groups, which has been initiated by L. A. Kurdachenko and J. Otal [KO 1, KO 2, KO 3], was a further natural prolongation of this research. This chapter is devoted to the main results of all mentioned above papers…

It is well known, that the relation “to be a normal subgroup” is not transitive. Therefore it is naturally to consider the groups, in which this relation is transitive…

The following sections are included:

• Bibliography

• Auther Index

• Subject Index