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Direct products and the Hopf property

Published online by Cambridge University Press:  09 April 2009

J. M. Tyrer Jones
J. M. Tyrer Jones
Affiliation:
New HallCambridge, England
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In [1] Neumann and Dey prove that the free product of two finitely generated Hopf groups is Hopf and ask whether a similar result holds for direct products. It is the purpose of this paper to show that this is not the case. We prove

THEOREM A. There exists a finitely generated group G satisfying the following conditions:

(i) G is isomorphic to a proper direct factor of itself;

(ii) G is the direct product of two Hopf groups.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 17 , Issue 2 , March 1974 , pp. 174 - 196
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Dey, I. M. S. and Neumann, Hanna, ‘The Hopf Property of Free Products’, Math. Z. 117, (1970), 325–339.CrossRefGoogle Scholar
[2]Camm, R., ‘Simple Free Products’, J. London Math. Soc. 28, (1953), 66–76.CrossRefGoogle Scholar
[3]Tyrer, J. M., ‘On Direct Products and the Hopf Property’, D. Phil. Thesis, University of Oxford, (1971).Google Scholar
[4]Magnus, W., Karrass, A. and Solitar, D., Combinatorial Group Theory, (Pure and Applied Mathematics, Vol. 13, Interscience Publishers, 1966).Google Scholar
[5]Neumann, B. H., ‘An Essay on Free Products of Groups with Amalgamations’, Philos, Trans. Roy. Soc. London, Ser. A. 246 (1954), 503–554.Google Scholar