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Sums of Complexes in Torsion-Free Abelian Groups

Published online by Cambridge University Press:  20 November 2018

J. D. Tarwater  and
R. C. Entringer
J. D. Tarwater
Affiliation:
Texas Technological College Lubbock, Texas 79409
R. C. Entringer
Affiliation:
University of New Mexico Albuquerque, New Mexico 87106
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The number of elements in the sum A + B of two complexes A and B of a group G which have multiple representations a + b = a '+ b' has been investigated by Scherk and Kemperman [1]. Kemperman [2] appealed to transfinite techniques (to order G) to prove:

If G is a torsion-free abelian group with finite subsets A and B with | B | ≥ 2, then at least two elements c of A + B admit exactly one representation c = a + b.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 4 , August 1969 , pp. 475 - 478
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Scherk, P. and Kemperman, J. H. B., Complexes in abelian groups. Canad. J. Math. 6 (1954) 230–237.Google Scholar
2. Kemperman, J. H. B., On complexes in a semigroup. Indag. Math. 18 (1956) 247–254.Google Scholar
3. Entringer, R. C., The 2Ω. property of torsion-free abelian groups. Amer. Math. Monthly 74 (1967) 301–302.Google Scholar
4. Fuchs, L., Abelian groups. (Pergamon Press, Oxford, 1960.)Google Scholar