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].

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.