Splittings of Abelian groups by integers (Q5903784)

Let G be a finite abelian group and M a subset of non-zero integers. M is said to split G if for some \(S\subset G\), every \(g\neq 0\) in G can be written uniquely in the form \(g=ms\) with \(m\in M\), \(s\in S\), and 0 is not of such a form. Splittings of groups arise in a geometric context [cf. \textit{W. Hamaker}, Aequationes Math. 9, 145-149 (1973; Zbl 0288.20072); \textit{S. Stein}, Am. Math. Mon. 81, 445-462 (1974; Zbl 0284.20048)]. The splitting is called non-singular, if \((m,| G|)=1\) for each \(m\in M\). Otherwise it is called singular. The splitting is called purely singular, if each prime that divides \(| G|\) divides at least one \(m\in M.\) Hickerson has conjectured that if the set \(M=\{1,2,...,k\}\) splits a finite abelian group G purely singularly, then G must be either C(1), \(C(k+1)\), or \(C(2k+1)\), where C(r) denotes the cyclic group of order r. And Hickerson proved the conjecture for \(k\leq 700\) with exceptions \(k=24\), 60, 62, 84, 144, 171, 180, 264, 312, 420, 480, 665, and 684, but this result is unpublished. Stein asked that if the set \(M=\{1,2,...,k\}\) splits \(G\times C(r)\) for a finite abelian group G, whether M splits C(r) when \(r\equiv 1 (mod k)\) [cf. \textit{S. Stein}, Rocky Mt. J. Math. 16, 277- 321 (1986; Zbl 0603.52007)]. In this paper, the author proves that if the set \(M=\{1,2,...,k\}\) splits \(G_ 1=G\times C(r)\) where G is a finite abelian group, \(r\equiv 1 (mod k)\), and k satisfies Hickerson's conjecture, then M splits C(r). Thus in view of the result of Hickerson, the author obtains directly the following assertion: for the set \(M=\{1,2,....,k\}\) with \(k\leq 700\) and \(k\neq 24\), 60, 62, 84, 144, 171, 180, 264, 312, 420, 480, 665, 684, if M splits \(G\times C(r)\) for a finite abelian group G, with \(r\equiv 1 (mod k)\), then M splits C(r). This gives a partial answer to Stein's question.