Hajós' theorem for multiple factorizations
Let \(G\) be a finite abelian group written multiplicatively with identity element \(e\). Let \(A_ 1, \dots, A_ n\) be subsets of \(G\). If each element \(g\) of \(G\) is expressible in precisely \(k\) ways in the form \(g = a_ 1 \dots a_ n\), \(a_ 1 \in A_ 1, \dots, a_ n \in A_ n\), then we say that the product \(A_ 1 \dots A_ n\) is a \(k\)-factorization of \(G\). When the product \(A_ 1 \dots A_ n\) is direct than it is a 1- factorization of \(G\) and will be called simply a factorization of \(G\). The subset \(A\) of \(G\) is defined to be cyclic if it is of the form \(\{e, a, a^ 2, \dots, a^{r-1}\}\). \textit{G. Hajós} proved that in every factorization of a finite abelian group by cyclic subsets at least one subgroup must occur among the factors [Math. Z. 47, 427-467 (1941; Zbl 0025.25401)]. The authors first give examples which show that Hajós' theorem does not extend to multiple factorizations, and then prove the following theorem: In a multiple factorization of a finite abelian group by cyclic subsets a subgroup always occurs among the factors, if the multiplicity is relatively prime to the order of the group.
scientific article; zbMATH DE number 638233