Groups with Many FC-Subgroups

✍ Scribed by Silvana Franciosi; Francesco de Giovanni; Yaroslav P. Sysak

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 120 KB
• Volume: 218
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7836

✦ Synopsis

DEDICATED TO DEREK J. S. ROBINSON ON THE OCCASION OF HIS 60TH BIRTHDAY

1. Introduction

A group G is called an FC-group if every element x of G has only Ž . finitely many conjugates in G, that is, if the centralizer C x has finite G index in G. There exists a wide literature on this subject, and the w x monograph 18 can be used as a general reference. In the last few years many authors have studied the structure of minimal-non-FC groups, i.e., those groups which are not FC-groups while all their proper subgroups Ž w x have the property FC see, for instance, 2, 4, 5, 10 and the last section of w x. 18 . Clearly Tarski groups are minimal-non-FC, and hence in this investigation it is necessary to impose some additional condition in order to avoid such groups. In the above mentioned articles, it has been proved that minimal-non-FC groups having proper commutator subgroup are Cernikov groups, and that every perfect locally graded minimal-non-FC group is a * This research was done while the last author was a visiting professor at the Universita di Ǹapoli ''Federico II'' supported by the ''Istituto Nazionale di Alta Matematica.'' He is grateful to the Department of Mathematics for its excellent hospitality.