Counting the Periodic Groups Generated by Two Finite Groups

In [6] Hickin and Phillips prove the existence of 2 non-isomorphic groups of exponent p for each sufficiently large prime p. The proof uses wreath products and embedding methods and relies on the existence of the infinite p-groups of Novikov and Adjan [7]. Phillips [8] had previously proved the existence of 2° non-isomorphic twogenerator p-groups for each prime p. The main theorem of this paper, proved by a completely different method, is a further result of this type.