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The Herzog-Schönheim Conjecture for Finite Nilpotent Groups

Published online by Cambridge University Press:  20 November 2018

Marc A. Berger ,
Alexander Felzenbaum  and
Aviezri Fraenkel
Marc A. Berger
Affiliation:
Department of Mathematics, The Weizmann Institute of ScienceRehovot 76100, Israel
Alexander Felzenbaum
Affiliation:
Department of Mathematics, The Weizmann Institute of ScienceRehovot 76100, Israel
Aviezri Fraenkel
Affiliation:
Department of Mathematics, The Weizmann Institute of ScienceRehovot 76100, Israel
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Abstract

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The purpose of this note is to prove the Herzog-Schônheim [3] conjecture for finite nilpotent groups. This conjecture states that any nontrivial partition of a group into cosets must contain two cosets of the same index (Corollary IV below). See Porubský [4, Section 8] for a perspective on coset partitions.

Keywords

05A1710A4520DI520D60coset partitionEuler φ-functionnilpotent group
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 3 , 01 September 1986 , pp. 329 - 333
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Berger M., A., Felzenbaum, A. and Fraenkel, A., Lattice parallelepipeds and disjoint covering systems, Discrete Math, (in press).Google Scholar
2. Burshtein, N., On natural covering systems of congruences having moduli occurring at most M times, Discrete Math. 14(1976), pp. 205–214.Google Scholar
3. Herzog, M. and Schônheim, J., Research problem No. 9, Canad. Math. Bull. 17(1974), p. 150 Google Scholar
4. Porubský, S., Results and problems on covering systems of residue classes, Mitteilungen aus dem Math. Sem. Giessen, Heft 150, Giessen Univ., 1981.Google Scholar
5. Rotman J., J., The Theory of Groups: An Introduction, Allyn and Bacon, Boston, 1973.Google Scholar