A proof of purely singular splitting conjecture

TL;DR

This paper proves Woldar's 1995 conjecture that only specific cyclic groups admit purely singular splittings by the set {1,2,...,k}.

Contribution

It provides a complete proof of the long-standing conjecture on purely singular splittings of finite abelian groups.

Findings

Confirmed that only cyclic groups of orders 1, k+1, and 2k+1 admit such splittings.

Established the necessary and sufficient conditions for purely singular splitting.

Resolved a 28-year-old conjecture in group theory.

Abstract

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 $|G|$ 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.