On the Groups of Britton's Theorem A

Abstract

In his simple proof of the unsolvability of the word problem for groups [1], J. L. Britton proved a normal form theorem (Britton's Lemma) for groups obtained by the HNN construction. In the appendix to that paper he described a generalization of the HNN construction and sketched the proof of a generalization of Britton's Lemma for this new construction, Britton's Theorem A. In this note we demonstrate that all groups obtained by means of this generalized construction are in fact HNN groups; and it will follow that Theorem A is simply a restatement of Britton's Lemma. This argument makes it clear that while Theorem A can be (and has been) useful in various group theoretic situations, in practice, every application of this generalized construction can be replaced by a straightforward application of the HNN construction.