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# A Reduction Theorem on Purely Singular Splittings of Cyclic Groups

Andrew J. Woldar
Proceedings of the American Mathematical Society
Vol. 123, No. 10 (Oct., 1995), pp. 2955-2959
DOI: 10.2307/2160647
Stable URL: http://www.jstor.org/stable/2160647
Page Count: 5
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 |G| 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 $\operatorname{gcd}(|G|, 6) = 1$.