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On the Groups of Britton's Theorem A

Published online by Cambridge University Press:  20 November 2018

George S. Sacerdote
George S. Sacerdote*
Affiliation:
Amherst College, Amherst, Massachusetts
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 3 , 01 June 1976 , pp. 635 - 639
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Britton, J. L., The word problem, Ann. Math. 77 (1963), 16–32.Google Scholar
2. Higman, G., Neumann, B. H., and Neumann, H., An embedding theorem for groups, J. London Math. Soc, 24 (1949), 247–254.Google Scholar
3. Miller, C. F. III, On Buttons Theorem A, Proc. Amer. Math. Soc, 19 (1968), 1151–1154.Google Scholar