Finite Groups with Nilpotent Centralizers

Abstract

Introduction.The purpose of this paper is to clarify the structure of finite groups satisfying the following condition:(CN) : the centralizer of any nonidentity element is nilpotent.Throughout this investigation we consider only groups of finite order.A group is called a (P)-group if it satisfies a group theoretical property (P).In this paper we shall clarify the structure of nonsolvable (CN)-groups and classify them as far as possible.This goal has been attained in a sense which we shall explain later.If we replace in (CN) the assumption of nilpotency by being abelian we get a stronger condition (CA).The structure of (CA)-groups has been known.In fact after an initial attempt by K. A. Fowler in his thesis [8], Wall and the author have shown that a nonsolvable (CA)-group of even order is isomorphic with LF(2, q) lor some q = 2n>2.A few years later the author [12] has succeeded in proving a particular case of Burnside's conjecture for (CA)groups, namely a nonsolvable (CA)-group has an even order.Quite recently Feit, M. Hall and Thompson [7] have proved the Burnside's conjecture for (CN)-groups.We can therefore consider groups of even order and focus our attention to the centralizers of involutions.We consider the condition (CIT):(CIT) : a group is of even order and the centralizer of any involution is a 2-group.There is no apparent connection between the class of (CN)-groups and the class of (CIT)-groups.But a nonsolvable (CN)-group is a (CIT)-group (Theorem 4 in Part I).This theorem reduces the study of nonsolvable (CN)groups to that of (CIT)-groups.Both properties (CN) and (CIT) are obviously hereditary to subgroups (provided that we consider only subgroups of even order in the case of (CIT)).Although it is true that a homomorphic image of a (CN)-group is also a (CN)-group (this statement is false for infinite groups), it is not an obvious statement.On the other hand it is not difficult to show that a factor group of a (CIT)-group is a (CIT)-group, provided that the order is even.This is due to the following characterization of (CIT)-groups : namely a (CIT)-group is a group of even order containing no element of order 2p with p>2 and vice versa.This makes the study of (CIT)-groups somewhat easier.The large part of this paper concerns the structure of (CIT)groups.