ON AMALGAMATED FREE-PRODUCTS OF RESIDUALLY P-FINITE GROUPS

We find a necessary and sufficient condition for an amalgamated free product of arbitrarily many isomorphic residually p-finite groups to be residually p-finite. We also prove that this condition is sufficient for a free product of any finite number of residually p-finite groups, amalgamating a cyclic subgroup, to be residually p-finite. We observe that a group is potent, if it is residually p-finite, for all primes p. Using this fact, we prove that a free product of finitely many groups, amalgamating a maximal cyclic subgroup is potent, if each factor is either free or finitely generated, torsion-free, and nilpotent. © 1993 Academic Press, Inc.