Conjugate separability in polycyclic groups

Abstract

A group G is said to be conjugate separable if, whenever x and y are elements of G which are not conjugate, there is a finite homomorphic image of G in which the images of x and y are not conjugate. The main result of this paper is that polycyclic-by-finite groups are conjugate separable. Conjugate separability was introduced by Blackburn [2], who showed that finitely generated torsion-free nilpotent groups are conjugate separable. This was extended to supersolvable groups by Kargapolov [5], and to finitely generated nilpotent-by-finite groups by Toh [8]. Remeslennikov [6] obtained the same result as this paper. His proof contained a number of mistakes, but Hartley’s review [MR43, #6313] pointed out that these could be corrected. This paper is taken from my Ph.D. thesis, Rice University, 1970, written under the direction of Steve Gersten. I did not become aware of Remeslennikov’s proof until after I has left Rice University. Both of our proofs are based on a number theoretic result of Chevalley [3]. The main difference between them is that Remeslennikov reduces directly to Chevalley’s result while I first extend Chevalley’s result to certain matrix groups, showing that they have the “congruence subgroup property.” The present form of the paper is a redraft, written in 1975, of a version submitted in April, 1970, and the following changes have been made. (1) Remeslennikov’s work has been acknowledged. (2) Some arguments have been shortened, and I have taken advantage of five years’ experience to improve the writing style. Here is an outline of the proof. Let