Splittings of Abelian groups by integers

Throughou t this pape r M will deno te a finite set of non-ze ro integers and G a finite Abe l i an group, wri t ten addit ively. Fo r m e M, g e G, mg denotes g + • • • + g, m t imes if m is posi t ive, and t m l g if m is negat ive, W e shall say that M s p l i t s G if there is a set S ~_ G such that each non-zero e lement in G can be expressed uniquely in the form m s where m e M and s e S and 0 is not of the form ms. M is a mul t ip l i e r set and S a spl i t t ing set. Spli t t ings of groups arise in a geometr ic context which we descr ibe briefly. Let k be a posi t ive integer and let M = {+1, ± 2 . . . . . ±k}. A s s u m e that M splits a group G of o rde r 2 k n + 1 with spli t t ing set S = {sl . . . . . s,}. Then it is poss ible to tile Eucl idean n -space with t ranslates of the cross fo rmed by a t taching an arm of length k on to each of the 2n faces of the unit cube. (The crosses consist of 2 k n + 1 unit cubes.) In fact the tiling consists of those crosses whose centers a re at the integer points (xl . . . . . x~) such that x l s l + • • • + x , sn = 0 in G. (See [3] or [7] for details.) In this pape r we shall cont inue the examina t ion of spl i t t ings begun in [8], [3], and [4], explor ing the exis tence or nonexis tence of spli t t ings for cer tain types of mul t ip l ie r sets M and groups G.