Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-07-06T09:14:43.451Z Has data issue: false hasContentIssue false
  • English
  • Français

On (k, l)-sets in cyclic groups of odd prime order

Published online by Cambridge University Press:  17 April 2009

T. Bier  and
A. Y. M. Chin
T. Bier
Affiliation:
Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia, e-mail: bier@mnt.math.um.edu.my, acym@mnt.math.um.edu.my
A. Y. M. Chin
Affiliation:
Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia, e-mail: bier@mnt.math.um.edu.my, acym@mnt.math.um.edu.my
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.

Let A be a finite Abelian group written additively. For two positive integers k, l with k ≠ l, we say that a subset S ⊂ A is of type (k, l) or is a (k, l) -set if the equation x1 + x2 + … + xk − xk+1−… − xk+1 = 0 has no solution in the set S. In this paper we determine the largest possible cardinality of a (k, l)-set of the cyclic group ℤP where p is an odd prime. We also determine the number of (k, l)-sets of ℤp which are in arithmetic progression and have maximum cardinality.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 63 , Issue 1 , February 2001 , pp. 115 - 121
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Mann, H.B., Addition theorems: The addition theorems of group theory and number theory, Interscience Tracts in Pure and Applied Mathematics 18 (John Wiley, New York, London, Sydney, 1965).Google Scholar
[2]Nathanson, M.B., Additive number theory: Inverse problems and the geometry of sumsets, Springer Graduate Texts in Mathematics 165 (Springer-Verlag, Berlin, Heidelberg, New York, 1996).CrossRefGoogle Scholar
[3]Street, A.P., ‘Sum free sets’, in Springer Lecture Notes in Mathematics 292 (Springer-Verlag, Berlin, Heidelberg, New York, 1972), pp. 123–272.Google Scholar
[4]Vosper, A.G., ‘The critical pairs of subsets of a group of prime order’, J. London Math. Soc. 31 (1956), 200–205.CrossRefGoogle Scholar
[5]Yap, H.P., ‘Maximal sum-free sets of group elements’, J. London Math. Soc. 44 (1969), 131–136.CrossRefGoogle Scholar