A proof of purely singular splitting conjecture

A set $M$ of nonzero integers is said to split a finite abelian group $G$ if there exists a subset $S\subseteq G$ such that $M\cdot S=G\setminus \{0\}$. Such a splitting is called purely singular if every prime divisor of $\mathrm{\mid }G\mathrm{\mid }$ divides some element of $M$. In 1995, Woldar \cite{W1995} conjectured that the finite abelian groups admitting a purely singular splitting by the set $\{1,2,\dots ,k\}$ are precisely the cyclic groups of orders $1$, $k+1$, and $2k+1$. In this paper, we prove this conjecture.