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A five lemma for free products of groups with amalgamations

Published online by Cambridge University Press:  17 April 2009

J. L. Dyer
J. L. Dyer
Affiliation:
Herbert H. Lehman College, City University of New York, and Institute for Advanced Study, Princeton, New Jersey, USA.
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Abstract

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This paper explores a five-lemma situation in the context of a free product of a family of groups with amalgamated subgroups (that is, a colimit of an appropriate diagram in the category of groups). In particular, for two families {Aα}, {Bα} of groups with amalgamated subgroups {Aαβ}, {Bαβ} and free products A, B we assume the existence of homomorphisms Aα → Bα whose restrictions Aαβ → Bαβ are isomorphisms and which induce an isomorphism A → B between the products. We show that the usual five-lemma conclusion is false, in that the morphisms Aα → Bα are in general neither monic nor epic. However, if all Bα → B are monic, Aα → Bα is always epic; and if Aα → A is monic, for all α, then Aα → Bα is an isomorphism.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 3 , Issue 1 , August 1970 , pp. 85 - 96
Copyright
Copyright © Australian Mathematical Society 1970

References

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