Factoring by subsets

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, aeA,beB.In 1942, G. Hajόs, 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,aeA, beB, and showed that under certain circumstances one of the sets is a group.While Hajόs's work grew out of a question concerning the partition of Euclidean ^-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, •••.&} or { + 1, ±2,, ±k}.The first three sections are algebraic and geometric, while the last two sections are number-theoretic.524 S. K. STEINsubset of G such that X = (A, B), we will say that B tiles X or that B is a complement of A (in X).For any subset T of G a set of the form Tg, geG is a riβf/ιt translate of T and a set of the form gT, geG is a ίβ/ί translate of Γ.Note that if G is a group and G = (A, B), then G = (Ac, τ/5) for any E and y in G. Thus if JB tiles G, so does τ/i?.For this reason we will assume when G = (A, B) and G is a group that both A and B contain the identity element of G.Note also that if G is a group and G = (A, B) and B = (C, Z)), then G = (AC, D).THEOREM 1.1.Let G be a groupoίd and G = {A, B), where B is finite and has β elements.If T czG has more than (k -l)β elements, then there is a right translate of A whose intersection with T has at least k elements.