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On the number of normal subgroups of an uncountable group

Part of: Foundations

Published online by Cambridge University Press:  09 April 2009

R. G. Burns ,
John Lawrence  and
Frank Okoh
R. G. Burns
Affiliation:
Department of Mathematics, York University, North York, Ontario, CanadaM3J 1P3
John Lawrence
Affiliation:
Pure Mathematics, University of Waterloo, Waterloo, OntarioCanadaN2L 3G1
Frank Okoh
Affiliation:
Department of Mathematics, Wayne State University Detroit, Michigan 48202, U.S.A.
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Abstract

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In this paper two theorems are proved that give a partial answer to a question posed by G. Behrendt and P. Neumann. Firstly, the existence of a group of cardinality ℵ1 with exactly ℵ1 normal subgroups, yet having a subgroup of index 2 with 2ℵ1 normal subgroups, is consistent with ZFC (the Zermelo-Fraenkel axioms for set theory together with the Axiom of Choice). Secondly, the statement “Every metabelian-by-finite group of cardinality ℵ1 has 2ℵ1 normal subgroups” is consistent with ZFC.

MSC classification

Secondary: 20A15: Applications of logic to group theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 3 , December 1986 , pp. 343 - 351
Copyright
Copyright © Australian Mathematical Society 1986

References

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