AUTHOR'S NOTE. This paper was written in 1980, but was not published at that time. The reference made to it in Miller's survey article [10], however, made me think that it might be worth publishing. It is unchanged except for the deletion of some remarks that either are outdated or no longer seem interesting. I am grateful to Professor Miller for his interest in the paper.

Introduction.

In the first part of this paper we give proofs of two embedding theorems for groups, and a version of the Adjan-Rabin construction for showing that many group-theoretic decision problems are unsolvable, which seem to be simpler than the standard ones. In the second part we consider some specific undecidability questions.

In [5], Higman, Neumann and Neumann proved the following theorem (see also [11]).

Theorem 1 Every countable group G can be embedded in a 2-generator group H. If G is n-relator, then H may be taken to be n-relator.

P. Hall (unpublished) proved that every countable group can be embedded in a finitely generated simple group. This was sharpened by Goryushkin [3] and Schupp [13] to

Theorem 2 Every countable group can be embedded in a 2-generator simple group.