PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 101, Number 4, December 1987 SOME PROBLEMS ON SPLITTINGS OF GROUPS. II SÁNDORSZABÓ (Communicated by Bhama Srinivasan) To Sherman Stein on his 60th birthday ABSTRACT. If G is an additive abelian group, S is a subset of G, M is a set of nonzero integers, and if each element of G\{0} is uniquely expressible in the form ms, where m € M and s € S, then we say that M splits G. A splitting is nonsingular if every element of M is relatively prime to the order of G; otherwise it is singular. In this paper we discuss the singular splittings of cyclic groups of prime power orders and the direct sum of isomorphic copies of groups. 1. Introduction. Let G be a finite abelian group written additively and let M be a set of nonzero integers. If there exists a subset S of G such that each element of G\{0} is uniquely expressible in the form ms, where m E M and s E S, then we say that the multiplier set M splits G with splitting set S and write G\{0} = MS. If |5| = n, then we say that M n-splits G. We will say that M n-packs G if the elements ms are distinct and nonzero, where m E M, s E S, and \S\ = n. Similarly, we will speak of an n-covering of G. Namely, M n-covers G if the elements ms cover G\{0}, where m E M, s E S, and \S\ = n. When the multiplier set has the form M = {1,... ,k} or M = {—k,...,— 1, 1,..., k}, questions about n-splitting, n-covering, and n-packing grow out of prob- lems on tiling, covering, and packing n-space by certain star bodies, the so-called crosses and semicrosses. See [7, 9, or 12], for instance. In this paper, which is a sequel to [14], we will examine packing and splitting of finite abelian groups. All groups will be assumed finite and abelian. 2. Packings. As in [12], g(k, n) and h(k, n) will denote the order of the smallest group which has an n-packing by M = {1,..., k} and M — {—k,..., —1,1,... ,k} respectively. Obviously, kn + 1 < g(n, k) and 2kn + 1 < h(k, n). The fact that the fractions (kn + l)/g(n,k) and (2kn + l)/h(k,n) equal the density of the densest integer lattice packing by crosses and semicrosses shows the significance of these functions. In [6] it was proved that, for n > 3, lim g(k,n)k~3/2 = 2cos(n/n), k—*oo and in [12] it was proved that, for n > 2, lim h(k,n)k~2 = 1. k—too Received by the editors March 15, 1986 and, in revised form, August 18, 1986. Presented at the International Group Theory Conference held in Debrecen (Hungary), August 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 20K01; Secondary 10E30. Key words and phrases. Splittings of finite abelian groups, factorizations of finite abelian groups, partitions of finite abelian groups, crosses, semicrosses, lattice tilings. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page 585 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use