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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A reduction theorem on purely singular splittings of cyclic groups


Author: Andrew J. Woldar
Journal: Proc. Amer. Math. Soc. 123 (1995), 2955-2959
MSC: Primary 20K01; Secondary 20D60
MathSciNet review: 1277139
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Abstract: A set M of nonzero integers is said to split a finite abelian group G if there is a subset S of G for which $ M \cdot S = G\backslash \{ 0\} $. If, moreover, each prime divisor of $ \vert G\vert$ divides an element of M, we call the splitting purely singular. It is conjectured that the only finite abelian groups which can be split by $ \{ 1, \ldots ,k\} $ in a purely singular manner are the cyclic groups of order $ 1,k + 1$ and $ 2k + 1$. We show that a proof of this conjecture can be reduced to a verification of the case $ \gcd (\vert G\vert,6) = 1$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1277139-2
PII: S 0002-9939(1995)1277139-2
Article copyright: © Copyright 1995 American Mathematical Society