Quasiperiodic Group Factorizations

Abstract.

If ${\mathcal{R}}$ is any ring or semi-ring (e.g., ${\mathbb{Z}}_+$) and G is a finite abelian group, two elements a, b of the group (semi-)ring ${\mathcal{R}}[G]$ are said to form a factorization of G if ab = rΣ _g∈G g for some $r \in {\mathcal{R}}$. A factorization is called quasiperiodic if there is some element g ∈ G of order m > 1 such that either a or b – say b – can be written as a sum b ₀ + ... + b _m−1 of m elements of ${\mathcal{R}}[G]$ such that ab _h = g ^h ab ₀ for h = 0, ... , m − 1. Hajós [5] conjectured that all factorizations are quasiperiodic when ${\mathcal{R}} = {\mathbb{Z}}_+$ and r = 1 but Sands [15] found a counterexample for the group $G = {\mathbb{Z}}_5 \times {\mathbb{Z}}_{25}$. Here we show however that all factorizations of abelian groups are quasiperiodic when ${\mathcal{R}} = {\mathbb{Q}}$ and that all factorizations of cyclic groups or of groups of the type ${\mathbb{Z}}_p \times {\mathbb{Z}}_p$ are quasiperiodic when ${\mathcal{R}} = {\mathbb{Z}}$. We also give some new examples of non-quasiperiodic factorizations with ${\mathcal{R}} = {\mathbb{Z}}_+$ for the smaller groups $G = ({\mathbb{Z}}_5)^2$ and $G = {\mathbb{Z}}_{35}$.