A reduction theorem on purely singular splittings of cyclic groups

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 $|G|$ 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 $\gcd (|G|,6) = 1$.