Simulated factorizations. II
A factorization of a finite abelian group is said to be simulated if it is obtained from a factorization into a direct product of subgroups by changing at most \(k\) elements in each subgroup. The question has been asked which values of \(k\) imply that in fact at least one subgroup must be left unaltered. This has been shown to be true for \(k=1\) but to be false, in general, for \(k=p-1\), where \(p\) is the least prime dividing the order of \(G\) [see part I, \textit{K. Corrádi}, \textit{A. D. Sands} and \textit{S. Szabó}, J. Algebra 151, No. 1, 12-25 (1992; Zbl 0774.20030)]. In the paper reviewed, it is shown to be true for \(k=p-2\). This result is stated as Theorem 5. Let \(G\) be a direct product of subsets \(A_ 1,\dots, A_ n\) and let there exist subgroups \(H_ 1,\dots, H_ n\) such that \(| A_ i|= | H_ i|\) and \(| A_ i|\leq | A_ i\cap H_ i|+p-2\), where \(p\) is the least prime factor of \(| G|\). Then there exists an \(i\) such that \(A_ i= H_ i\).
scientific article; zbMATH DE number 85813