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When are dense subgroups of LCA groups closed under intersections?

Published online by Cambridge University Press:  17 April 2009

M.A. Khan
M.A. Khan
Affiliation:
Department of Mathematical Sciences, Umm al-Qura University, PO Box 3711, Makkah, Saudi Arabia
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Abstract

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Let G be a nondiscrete locally compact Hausdorff abelian group. It is shown that if G contains an open torsion subgroup, then every proper dense subgroup of G is contained in a maximal subgroup; while if G has no open torsion subgroup, then it has a dense subgroup D such that G/D is algebraically isomorphic to R, the additive group of reals. With each G, containing an open torsion subgroup, we associate the least positive integer n such that the nth multiple of every discontinuous character of G is continuous. The following are proved equivalent for a nondiscrete locally compact abelian group G:

(1) The intersection of any two dense subgroups of G is dense in G.

(2) The intersection of all dense subgroups of G is dense in G.

(3) G contains an open torsion subgroup, and for each prime p dividing the positive integer associated with G, pG is either open or a proper dense subgroup of G.

Finally, we construct a locally compact abelian group G with infinitely many dense subgroups satisfying the three equivalent conditions stated above.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 49 , Issue 1 , February 1994 , pp. 59 - 67
Copyright
Copyright © Australian Mathematical Society 1994

References

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