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On a problem in the theory of ordered groups

Published online by Cambridge University Press:  17 April 2009

Colin D. Fox
Colin D. Fox
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Abstract

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The group G presented on two generators a, c with the single defining relation a−1c2a = c2a2c2 [proposed by B.H. Neumann in 1949 (unpublished), discussed by Gilbert Baumslag in Proc. Cambridge Philos. Soc. 55 (1959)] has been considered as a possible example of an orderable group which can not be embedded in a divisible orderable group, contrary to the conjecture that no such examples exist. It is known from Baumslag's discussion that G can not be embedded in any divisible orderable group. However, it is shown in this note that G is not orderable, and thus is not a counter-example to the conjecture.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 6 , Issue 3 , June 1972 , pp. 435 - 438
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Baumslag, Gilbert, “Wreath products and p–groups”, Proc. Cambridge Philos. Soc. 55 (1959), 224–231.CrossRefGoogle Scholar
[2]Fuchs, Lászlo, Teilweise geordnete algebraieche Strukturen (Akadémlai Kiadó, Budapest, 1966).Google Scholar
[3]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