The Hajós Factorization of Elementary 3-Groups

If G is a finite abelian group and n > 1 is an integer, we say that G has the Hajós n-property, or is n-good if from each decomposition G = S1S2…Sn of G into a direct product of subsets, it follows that at least one of the Si is periodic, meaning that there exists x G − {e} such that xSi = Si. Otherwise, G is said to be n-bad. In this paper, we show that if G is an elementary abelian 3-group of order 3n, then G is (n − 1)-good.