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On subgroups of amalgamated free products

Published online by Cambridge University Press:  24 October 2008

Edward T. Ordman
Edward T. Ordman
Affiliation:
University of Kentucky, Lexington, Kentucky, U.S.A.
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Extract

In 1934 Kurosh(9) proved that ‘a subgroup of a free product of groups is again a free product’. Many other proofs of this, and attempts to generalize it to amalgamated free products, have appeared (e.g. (7), (1), (10) and (8)). Recently the theory of groupoids has been applied to this area with increasing success. In 1966 Higgins (6) used groupoids to prove the generalization of Grushko's Theorem (3) due to Wagner (14).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 69 , Issue 1 , January 1971 , pp. 13 - 23
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

REFERENCES

(1)Baer, R. and Levi, F. W.Freie Produkte und ihre Untergruppen. Compositio Math. 3 (1936), 391–398.Google Scholar
(2)Brown, R.Elements of modern topology (McGraw-Hill; London, 1968).Google Scholar
(3)Grushko, I.On the bases of a free product of groups (Russian). Mat. Sb. 8 (1940), 169–182.Google Scholar
(4)Hall, , Marshall, Jr. The theory of groups (Macmillan; New York, 1959).Google Scholar
(5)Higgins, P. J.Presentations of groupoids, with applications to groups. Proc. Cambridge Philos. Soc. 60 (1964), 7–20.CrossRefGoogle Scholar
(6)Higgins, P. J.Gruskho's theorem. J. Algebra 4 (1966), 365–372.Google Scholar
(7)Kalashnikov, V. and Kurosh, A.Freie Produkte der Gruppen mit vereinigten Untergruppen der Zentren (Russian). Dokl. Akad. Nauk, SSSR, 6 (vol. 1, 1935), 285–286. Cited from Zbl. Math. 11 (1935), 151.Google Scholar
(8)Karrass, A. and Solitar, D. The subgroups of a free product of two groups with an amalgamated subgroup (to appear).Google Scholar
(9)Kurosh, A.Die Untergruppen der Freien Produkte von beliebigen Gruppen. Math. Ann. 109 (1934), 647–660.Google Scholar
(10)Neumann, , Hanna, Generalized free products with amalgamated subgroups. I. Amer. J. Math. 70 (1948), 590–625Google Scholar
II. Amer. J. Math. 71 (1949), 491–540.Google Scholar
(11)Ordman, E. T. Amalgamated free products of groupoids (Ph.D. thesis, Princeton University, 1969).Google Scholar
(12)Ordman, E. T.Subgroups of amalgamated free products. Bull. Amer. Math Soc. 76 (1970), 358–360.Google Scholar
(13)Stallings, J. R.A topological proof of Grushko's theorem on free products. Math. Z. 90 (1965), 1–8.Google Scholar
(14)Wagner, D. H.On free products of groups. Trans. Amer. Math. Soc. 84 (1957), 252–278.Google Scholar