Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-05T12:29:43.588Z Has data issue: false hasContentIssue false

A topological proof in group theory

Published online by Cambridge University Press:  24 October 2008

D. E. Cohen
Affiliation:
University of Birmingham

Extract

The topological theory of covering spaces may be used to prove results in group theory, for instance, the Kuros-Reidemeister-Schreier theorem (1). It seems likely that such methods can be applied to prove the Freiheitsatz (4) and the identity theorem (3), and also perhaps Lyndon's conjecture, that the normal closure in a free group F of a, single element r is freely generated by conjugates of r. However, although these problems may easily be stated in topological terms, no such proof is at present known. In this paper we prove a related result.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1963

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Baer, R., and Levi, F., Freie Produkte und ihrer Untergruppen. Compositio Math. 3 (1930), 391398.Google Scholar
(2)Hilton, P. J., and Wylie, S., Homology theory (Cambridge, 1960).CrossRefGoogle Scholar
(3)Lyndon, R. C., Dependence and independence in free groups. J. Reine Angew. Math. 210 (1962), 148174.CrossRefGoogle Scholar
(4)Magnus, W., Uber diskontinuierliche Gruppen mit einer definierenden Relation. J. Reine Angew. Math. 163 (1930), 141165.CrossRefGoogle Scholar
(5)Neumann, H., Generalised free products with amalgamated subgroups. II. American J. Math. 71 (1949), 491540.CrossRefGoogle Scholar