Vol. 22, No. 3, 1967

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ISSN: 0030-8730
Factoring by subsets

Sherman K. Stein

Vol. 22 (1967), No. 3, 523–541
Abstract

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.

Mathematical Subject Classification
Primary: 05.22
Secondary: 10.00
Milestones
Received: 6 April 1966
Published: 1 September 1967
Authors
Sherman K. Stein