Embedding Countable Groups in 2-Generator Groups

Abstract

The aim of this note is to popularize a simple proof, due to Neumann and Neumann [5], of the fact that every countable group is embeddable in a 2-generator group. This was first proved by Higman, Neumann, and Neumann [2, Theorem IV] and (independently) Freudenthal [2, p. 254], using free products with amalgamations. The proof given by Neumann and Neumann [5] used wreath products, which are widely regarded as no less terrifying than free products with amalgamations. I am indebted to Professors A. M. W. Glass, G. Higman, and P. M. Neumann, each of whom pointed out (in response to an earlier version of this note) that the Neumann-Neumann proof is really quite simple, and that it can easily be expressed directly in terms of permutations. Here, then, is a short proof that assumes no more background than is ileeded to understand the statement of the theorem. As usual, Z is the set of integers and NJ is the set of natural numbers; (a, b> is the group generated by a and b; Sym(fQ) is the group of all permutations of a set Ql; permutations are regarded as right operators, and are composed from left to right.