Abstract

A subgroup H of a group G is called pronormal in G if for each element g ∈ G the subgroups H and Hg are conjugate in 〈H, Hg〉. Pronormal subgroups have been introduced by P. Hall, and they play an important role in many studies dedicated to normal structure and Sylow theory of finite and infinite groups and in investigations of arrangement of subgroups in infinite linear groups over rings. Many interesting and important developments have been lately completed in this area by different authors. Thanks to these results, we can see that pronormal subgroups and some other types of subgroups related to them (such as contranormal, abnormal, polynormal, paranormal, permutable subgroups, and so on) are very closely connected to transitivity of some group properties (such as normality, permutability and other) and to (locally) nilpotency of a group. In the current survey, we try to reflect some important new results in this area.

Introduction

Recall that a subgroup H of a group G is called pronormal in G if for each element g ∈ G the subgroups H and Hg are conjugate in 〈H, Hg〉. Pronormal subgroups have been introduced by P. Hall. Important examples of pronormal subgroups are the Sylow p-subgroups of finite groups, the Sylow π-subgroups of finite soluble groups, the Carter subgroups of finite soluble groups and many others. Finite groups with all pronormal subgroups have been described by T.A. Peng in [41, 42].

Abstract

In this paper we present a short survey discussing some recent results in the theory of infinite dimensional linear groups.

Introduction

In this paper R will denote a ring, G will denote a group and A will be a right RG–module. When R is a field, we shall denote it by F and, of course, A is then also a vector space over F. The group GL(F,A), of all F-automorphisms of A, and its subgroups, are called linear groups. Linear groups have played a very important role in algebra and other branches of mathematics. If dim FA (the dimension of A over F) is finite, n say, then a subgroup G of GL(F, A) is called a finite dimensional linear group. It is well known that in this case GL(F,A) can be identified with the group of all invertible n × n matrices with entries in F. The subject of finite dimensional linear groups is among the most studied branches of mathematics, having been built using the interplay between algebraic, geometrical, combinatorial and other methods. This theory is rich in many interesting and important results.

However, the study of the subgroups of GL(F,A) in the case when A has infinite dimension over F has been much more limited and normally requires some additional restrictions. One natural type of restriction to use here is a finiteness condition. The most fruitful example of such restrictions to date has undoubtedly been that of finitary linear groups.

Abnormal and pronormal subgroups have appeared in the process of investigation of some important subgroups of finite (soluble) groups such as Sylow subgroups, Hall subgroups, system normalizers, and Carter subgroups. Let H be a subgroup of a group G. We recall that a subgroup H is abnormal in G if g ∈ 〈H, Hg〉 for each element g ∈ G; and a subgroup H is pronormal in G if for each element g ∈ G, H and Hg are conjugate in 〈H, Hg〉. Pronormal subgroups have been introduced by P. Hall in his lectures in Cambridge; he also introduced abnormal subgroups in his paper, whereas the term abnormal comes from R. Carter. These subgroups and their generalizations have shown to be very useful in the finite group theory. It appears to be logical to employ such fruitful concepts in infinite groups. However, in some classes of infinite groups these mentioned subgroups gain such properties that they cannot posses in the finite case. For example, it is well-known that every finite p-group has no proper abnormal subgroups. Nevertheless, A. Yu. Olshanskii has constructed a series of impressive examples of infinite finitely generated p-groups saturated with abnormal subgroups. Concretely, for a large enough prime p there exists an infinite p-group G whose all proper subgroups have prime order p [18, Theorem 28.1]. In particular, every proper non-identity subgroup of G is maximal, and being non-normal, is abnormal.

Abstract

The authors report recent work concerning infinite dimensional linear groups with various finiteness conditions on subgroups of infinite central dimension.

Introduction

In this paper we give a survey of recent work of the authors and others. We recall that a group G that is isomorphic to a group of automorphisms of a vector space A over a field F is called a linear group and we denote the group of all such automorphisms by GL(F,A). If dimFA, the dimension of A over F, is finite, n say, then we say that G is a finite dimensional linear group and it is then well-known that GL(F,A) can be identified with the group of n × n matrices with entries in F. From the outset, finite dimensional linear groups have played an important role in mathematics. This is partly due to the correspondence mentioned above, but also because of the rich interplay between geometrical and algebraic ideas associated with such groups.

The study of the subgroups of GL(F,A) in the case when A is infinite dimensional over F has been much more limited and normally requires some additional restrictions. The circumstances here are similar to those present in the early development of Infinite Group Theory. One approach there consisted in the application of finiteness conditions to the study of infinite groups.

Abstract

This is a survey indicating the newest results on infinite-dimensional linear groups.

Let F be a field, A – a vector space over F. The group GL(F, A) of all automorphisms of A and its distinct subgroups (the linear groups) are the oldest subjects of investigation in Group Theory. Naturally, the first step here was the investigation of the case, when A has finite dimension over F. Under this condition every element of GL(F, A) (a non-singular linear transformation) defines some non-singular n × n - matrix over F where n = dimF A. Thus, for the finite-dimensional case the theory of linear groups coincides with the theory of matrix groups. That is why the theory of finite-dimensional linear groups is one of the best-developed theories in algebra.

The case when dimF A is infinite is totally different. The study of this case always requires some additional essential restrictions. The situation here is similar to that one which appeared in the initial period of development of Infinite Group Theory. One of the very fruitful approaches is associated with the finiteness conditions. It seems very meaningful to apply it to the linear groups.

Take into consideration the following important aspect. Many problems related to groups with finiteness conditions require the study of modules over some group rings, frequently the group rings of the type FG where F is a field. Thus, the respectively finiteness conditions from groups could be transferred on the appropriate modules.

Abstract

In the present paper we start to study the soluble groups in which every non-subnormal subgroup has finite special rank.

Introduction

Let G be a group, Lnon–sn(G) the set of all non-subnormal subgroups of G. If Lnon–sn(G) = Ø then we obtain a group, every subgroup of which is subnormal. The study of these groups was very fruitful and has brought many interesting results (see the books [12], [13], [8]). The groups G in which the set Lnon–sn(G) is “very small” in some sense is the natural next consideration. For many domains of Infinite Group Theory “to be very small” means to satisfy some finiteness conditions. The first natural finiteness conditions in Group Theory were the classical minimal and maximal conditions. The groups with minimal condition for non-subnormal subgroups have been considered by S. Franciosi and F. de Giovanni [1]. The groups with dual maximal condition for non-subnormal subgroups have been studied by L.A. Kurdachenko and H. Smith [4]. The groups with maximal condition for non-subnormal subgroups is nearly allied to the groups, in which the set Lnon–sn(G) consists only of the finitely generated subgroups. Such groups have been considered. The results of papers [1], [4] have been extended on groups with weak minimal and maximal condition on non-subnormal subgroups [5], [6]. The weak minimal and maximal conditions are connected with the concept of special rank (or Mal'tsev-Prüfer rank).