The partition problem for finite abelian groups

AbstractBy the aid of a slight generalization of the Hales-Jewett theorem [Trans. Amer. Math. Soc. 106 (1963), 222–229] we investigate the partition problem for finite Abelian groups. In particular the partition problem for the class of finitely generated free modules over Zq is solved. By the results of Deuber and Rothschild [“Coll. Math. Soc. János Bolyai 18,” 1976] this yields a complete characterization of those finite Abelian groups with respect to which the class FAB of all finite Abelian groups has the partition property. Especially it turns out that FAB has the partition property with respect to the cyclic group Zm, m > 1