In this note a simple principle is explained for constructing a transformation group which is a free product of given transformation groups. The principle does not seem to have been formulated explicitly, though it has been used in a more or less vague form in the theory of discontinuous groups (see, for instance, L. R. Ford, Automorphic functions, vol. I, pp. 56–59). It is perhaps of interest that the formulation given here is purely set-theoretic, without any topology, and that it can apply to any free product, whatever the cardinal number of the set of factors. The principle is used to establish the closure under the formation of countable free product of the family of groups which can be represented as discontinuous subgroups of a certain group of rational projective transformations. (The word ‘discontinuous’ is used here in a weak sense, defined later.) Finally, these results are applied to give a new proof of the theorem of Gruenberg that a free product of residually finite groups is itself residually finite (K. W. Gruenberg: Residual properties of groups, Proc. London Math. Soc. (3), 7 (1957), 29–62. See Corollary (ii) of Theorem 4.1, p. 44). The present proof is completely different from Gruenberg's and seems to be of interest for its own sake, though it does not appear to lead to Gruenberg's other results in this connexion. I have to thank Mr W. J. Harvey and Mr C. Maclachlan for checking a first draft of this paper and pointing out a few errors.
The free product of residually finite groups amalgamated along retracts is residually finite