Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T03:09:47.114Z Has data issue: false hasContentIssue false
  • English
  • Français

A question of Babai on groups

Published online by Cambridge University Press:  17 April 2009

J.L. Hickman  and
B.H. Neumann
J.L. Hickman
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT
B.H. Neumann
Affiliation:
Division of Mathematics and Statistics, Commonwealth Scientific and Industrial Research Organization, Canberra, ACT.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

László Babai raised the question whether every infinite group G contains a set B of elements, of cardinal equal to the order of G, such that for elements a, b, c of B the equation a = bc−1b implies a = c. We show that the answer is affirmative for certain classes of groups, including the soluble groups and the countable groups, if the Axiom of Choice is assumed, but negative, even for abelian groups, if a different axiom, compatible with Zermelo-Fraenkel set theory but incompatible with the Axiom of Choice, is assumed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 13 , Issue 3 , December 1975 , pp. 355 - 368
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Adyan, S.I., “Periodic groups of odd exponent”, Proc. Second Internat. Conf. Theory of Groups, Austral. National Univ. 1973, 8–12 (Lecture Notes in Mathematics, 372. Springer-Verlag, Berlin, Heidelberg, New York, 1974).CrossRefGoogle Scholar
[2]Babai, László, in litt.Google Scholar
[3]Gödel, Kurt, The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory (Annals of Mathematics Studies, 3. Princeton University Press, Princeton, New Jersey, 1940; 7th printing, 1966).Google Scholar
[4]Hickman, John Llewellyn, “An investigation of several finiteness criteria relative to Zermelo-Fraenkel set theory” (PhD thesis, Flinders University of South Australia, 1971). See also: Abstract, Bull. Austral. Math. Soc. 7 (1972), 300.CrossRefGoogle Scholar
[5]Monro, G.P., “Some independence results for weak axioms of choice” (PhD thesis, University of Bristol, 1971).Google Scholar
[6]Monro, G.P., “Small sets with large power sets”, Bull. Austral. Math. Soc. 8 (1973), 413–421.CrossRefGoogle Scholar