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If a group G is the direct
product of two of its subgroups, A and B, then every element of G is uniquely
expressible in the form ab,a ∈ A,b ∈ B. In 1942, G. Hajós, in order to solve a
geometric problem posed by Minkowski, introduced the notion of the direct product
of subsets. He said that the group G is the direct product of two of its subsets, A
and B, if each element of G is uniquely expressible in the form ab,a ∈ A,
b ∈ B, and showed that under certain circumstances one of the sets is a
group.
While Hajós’s work grew out of a question concerning the partition of Euclidean
n-dimensional space into congruent cubes, the present paper grew out of a question
concerning partitions into congruent “crosses” and is concerned primarily with the
existence of factorizations of the semigroup of integers modulo m into subsets A
and B, of which A is prescibed as {1,2,⋯,k} or {±1,±2,⋯,±k}. The first
three sections are algebraic and geometric, while the last two sections are
number-theoretic.
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