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Let G be a group, written
additively, M a set of integers, and S a subset of G. We will say that M
and S form a splitting of G if every nonzero element of G has a unique
representation as a product ms with m ∈ M and s ∈ S, while 0 has no such
representation. (Here “ms” denotes the sum of ms’s if m ≥ 0 and denotes
−((−m)s) if m < 0.) Splittings arise in connection with the problem of tiling
Euclidean space by translates of certain unions of unit cubes, called “crosses” and
“semicrosses”.
In this paper, we develop a counting technique which gives information about S if
M and G are known. This technique is used to reduce the study of splittings of finite
abelian groups to those of nonsingular splittings and of purely singular splittings. (A
splitting is nonsingular if every element of M is relatively prime to |G|; it is purely
singular if, for every prime divisor p of |G|, some element of M is divisible by
p.) Next, it is shown that every splitting of a noncyclic abelian p-group is
nonsingular. A construction is then given which yields many purely singular
splittings.
We then discuss a number of results and examples, including some infinite and
nonabelian groups, and close with a list of open problems.
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